The first half was one to forget for the New England Patriots. But the final two quarters showed vast improvement.

"Just to keep our composure and chip away at it little by little," Patriots coach Mike Vrabel said of his message at halftime. "Get connected and figure out a way to complement each other. Get some points, get some stops, get some turnovers."

And with that, the Patriots picked up their fifth win of the season and third against a divisional opponent, defeating the Miami Dolphins 24-19.

Here are the key takeaways from Sunday's game:

1. Kevin Byard makes Malik Willis pay, and DeMario Douglas cashes in

In the fourth quarter, on the ensuing drive after the Patriots reclaimed the lead, Malik Willis sent a pass over the middle, but it was tipped by DaQuan Jones and fell right into the waiting arms of safety Kevin Byard, who (finally) got his first interception as a Patriot.

"Happy for KB," Christian Gonzalez said after the game. "He got his first one of the year. It was good. It was good momentum. We talked about it all week. We wanted to come in and break that drought. Get some picks, get some interceptions."

Then, New England took no time in taking advantage of the opportunity. Rhamondre Stevenson began the drive with a five-yard run, but then it was Maye who used his legs to buy time and extend the play, ultimately finding DeMario Douglas. The receiver then used his speedy nature and ability to cause defenders to miss, dodging one tackle and then splitting the defenders to finish it off for a 48-yard score.

With the made extra point by Borregales, it marked 21 unanswered points by the Patriots in roughly 12 minutes.

2. One last chance by Miami

The Dolphins got back in the game with just over two minutes remaining, as Willis hit Kevin Coleman for a 35-yard touchdown. Cornerback Charles Woods was in coverage, but appeared to have slipped, which left Coleman wide open and Willis took advantage. They then went for a two-point conversion, but it was unsuccessful.

Afterwards, Miami lined up for an onside kick, but Stevenson fielded the ball, giving New England a chance to put it away. And then, New England managed to get one final first down following the two-minute warning, which then resulted in a victory formation.

3. Drake Maye continues to impress

Maye has been able to thrive in recent weeks despite having a very depleted receiving corps that only continues to get even more banged up. The quarterback finished Sunday's game going 16-of-20 for 199 passing yards and two passing touchdowns, as well as eight carries for five yards and a rushing touchdown.

After the game, Vrabel praised the quarterback's ability to overcome the injuries, also highlighting that it's his job and he's doing it well.

"That's what being a great quarterback in the National Football League is," Vrabel said. "Not a whole lot of excuses. You go out there and find ways to win for your football team. I know that's who he is."

4. Kyle Williams gets involved

After going three-and-out to start the second half, Drake Maye came back and fired off a big pass to Kyle Williams, hitting the receiver for a 43-yard score. After Williams crossed the line, the ball came loose and for some reason, the officials ruled it as incomplete.

Mike Vrabel was heated with the call, launching the red challenge flag so far that the broadcast even said he had a "better throwing day than Baker Mayfield." And as expected, the call was overturned, with it being ruled a touchdown. Andy Borregales made the extra point, putting New England back within three.

"I was a lot more certain than the officials were the first time they called it," Vrabel said of his decision to throw the challenge flag. "I guess I'd say that."

5. A risky decision by Mike Vrabel

After Williams' score, New England remained down by three following a nice defensive stop that was aided by a taunting penalty called against Ollie Gordon.

With the ball just outside Miami's 20-yard line, Vrabel opted to keep the offense out there on 4th-and-2 following an interesting third-down play call rather than taking the three points and tying it up. Maye hit DeMario Douglas, who was initially ruled short. But because it was a turnover on downs, the play was automatically reviewed, and the call on the field was overturned, with the Patriots getting the first down.

This marked their second successful fourth-down conversion of the game.

6. Patriots' secondary puts on inexcusable first-half performance

For the first time since Week 1, the Patriots had a fully healthy secondary, but it certainly didn't look like that in the first half. They were getting beaten by a 1-6 Miami offense, with Malik Willis going 12-of-15 for 115 yards. His top receiver was Malik Washington, who caught four of his five targets for 35 yards and a touchdown.

"First half, we weren't really getting off the field on third down," Gonzalez said. "We started off hot, and then I think after the first the third drive, they started driving. So I mean, just getting off the field, getting the ball to Drake and the offense, and letting them do their thing."

7. From bad to worse

A brutal drive for the Patriots left the team with no points and without Drake Maye's top target.

After Zane Gonzalez nailed a 52-yard field goal, with a Doink that ended up still falling through the uprights. Then, the Patriots came out trailing the Dolphins in a game where they were expected to be dominating, even with all the injuries. And things just got worse.

On the first play, Maye hit Hunter Henry for a gain of four yards, but the tight end was shaken up after the play, went into the blue medical tent, and then departed for the locker room. He was listed as questionable to return with a foot injury, but in the fourth quarter, he was downgraded to out.

The Patriots then began to rely heavily on the run, with Rhamondre Stevenson and Lan Larison getting some snaps. But on 3rd-and-1, Eli Raridon was flagged for a false start, pushing them back five yards. Maye then hit Stevenson for a gain of two, bringing up fourth down, and head coach Mike Vrabel made the decision to keep the offense out.

They converted, with Maye hitting Efton Chism for a gain of 12 yards. However, they were unable to pick up another first down, with some small runs and a bad snap by Jared Wilson. Instead of going for it again, Vrabel sent out the field goal unit, but Andy Borregales missed the 40-yard attempt.

8. Efton Chism coughs up the ball in the worst spot

After Maye hit Efton Chism, the second-year receiver tried to run with it to get some extra yards, but the ball was punched out, and the Dolphins recovered, giving them field position on New England's 19-yard line.

However, the defense came up big once again. Ollie Gordon had a gain of no yards on the first run attempt, and then on his second, it was Christian Elliss and Cory Durden who blew up the attempt for a loss of three yards. Malik Willis then hit Chris Bell for a gain of nine yards, and Miami head coach Jeff Hafley decided to send out Zane Gonzalez, who made the 31-yard field goal to tie things up at three with just over three minutes remaining in the first quarter.

With this fumble, the Patriots haven't had a turnover-free game all season, as it marked their 15th turnover of the year.

And things didn't get much better for New England, as they went three-and-out on the following drive, which included former Patriots pass rusher Josh Uche beating left guard Walter Rouse and sacking Maye for a loss of 12.

9. Cory Durden gets first career sack

Cory Durden recorded his first career sack on the Dolphins' opening drive, pressuring quarterback Malik Willis and forcing him to slide before the line of scrimmage for a two-yard loss.

On the following drive, Gabe Jacas and Leonard Taylor put pressure on Willis on the rollout, with Durden closing in and finishing the play with another sack. After playing 32 games without recording a sack, it took him only seven minutes in Sunday's game to record 1.5 sacks.

"It meant a lot. Obviously, I don't want to be known as only a nose (tackle)," Durden said. "I feel like I can rush the passer and get to the quarterback. So it's nice to finally see some production in that category."

In the team's first two defensive series, they recorded a total of three sacks, with Jacas, Elliss, and Christian Barmore each recording half a sack as well. And with that, the Dolphins went three-and-out both times.

Durden finished the game with four total tackles, 1.5 sacks, and two tackles for loss, along with two quarterback hits.

10. Other notes

• Patriots linebacker Christian Elliss was today's game-day captain. The Dolphins won the toss and elected to receive.

• Corey Kiner and Lan Larison served as the Patriots' kick returners.

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