EAST RUTHERFORD, N.J. — The sight of New York Giants fans streaming for the exits with 11:31 remaining in Sunday’s game was as good a sign as any that the New Orleans Saints received the message.

What transpired after that moment proved they may still have a way to go.

Days after owner Gayle Benson publicly criticized her team’s recent performance in the midst of a three-game losing streak, the Saints built a huge lead and then survived a furious fourth-quarter rally to beat the New York Giants 34-27. At one point, the Saints went on a 34-0 run to take a 34-10 lead, then held as the Giants scored the game’s final 17 points behind backup quarterback J.J. McCarthy.

McCarthy, who threw for a whopping 243 yards in the fourth quarter, fired incomplete on a fourth-and-1 from the Saints’ 14-yard line to seal the outcome.

“You’re never going to apologize for winning in the NFL,” quarterback Tyler Shough said. “Doing that as a team felt really good. Our process was there, our play style was there, (but) we’ve got to put that away earlier as a team and we know that.”

To snap their skid, the Saints (2-4) found answers that have eluded them in recent weeks.

The run game finally found its footing, gashing the Giants for 175 yards on the ground. New Orleans mostly took care of the football when on offense, and the defense made New York pay for its mistakes by forcing four turnovers of its own. A special-teams unit that has been performing poorly made huge contributions in the kicking game, return game and in coverage.

“We talked about the ball going into the game; we created four turnovers, we gave the one away which was a critical one,” coach Kellen Moore said. “We talked about the physicality we have to play with, that was an objective of ours, and to rush for 175 (yards) was an important part for us.”

To get there, the Saints needed a jolt out of their regular early-game struggles.

The Saints’ first five drives netted just 55 yards of offense. Shough was erratic, completing just seven of his first 18 passes. Penalties torpedoed one drive when backup left tackle Asim Richards was penalized for holding on consecutive plays. Kicker Daniel Carlson missed a 55-yard field goal at the end of the Saints’ most promising drive.

Even when the Saints finally got on the scoreboard, it felt like a missed opportunity: Rookie Barion Brown returned a kick 41 yards to set up a short field near the end of the half, but New Orleans to settle for a 58-yard Carlson field goal with 1:18 remaining in the first half.

The spark the team desperately needed came in the form of special-teams ace Isaiah Stalbird on the ensuing kickoff, when he met Giants kick returner Charlie Jones like a heat-seeking missile.

Stalbird separated Jones from the ball, and rookie tight end Oscar Delp recovered the loose ball. It was as if a dam broke free after that play, with the New Orleans offense roaring to life to seize control of the game.

Shough ripped off a 25-yard run on the first play after the fumble recovery, setting New Orleans up at the goal line, where he would eventually find Bryce Lance for a game-tying touchdown pass on fourth-and-goal.

That was the start of three consecutive touchdown drives for the Saints, who scored on six of seven drives in the middle portion of the game to take a commanding lead.

They got contributions from Alvin Kamara, who looked like he’d turned back the clock making three defenders miss on a 31-yard touchdown catch. Shough took one in on his own from a yard out, then looked like he iced the game with an 18-yard scoring strike to Chris Olave. The defense also pitched in, sacking Giants starter Jameis Winston four times and picking him off twice.

Winston’s final interception set up the Olave score that gave the Saints a 34-10 lead. The fans began to leave the game, and Moore yanked some of his starters. The Giants looked like they were doing the same, inserting McCarthy into the lineup.

The game was far from over.

McCarthy threw an interception of his own, but he stayed aggressive, connecting on several explosive passes against the Saints defense, including gains of 49 to Jones and 46 to former LSU star Malik Nabers, who finished with 165 yards on 12 catches. Running back Tyrone Tracy started the comeback with a 5-yard touchdown run on third-and-2.

New Orleans put its offensive starters back in, but the Saints were starting to get overwhelmed. McCarthy closed the gap further with a 24-yard touchdown pass to tight end Isaiah Likely. New York decided to cut the lead to seven by kicking a field goal on a fourth-and-6 play, trusting the defense to get another stop.

New York had an opportunity to tie the game or take the lead when it forced the Saints to go three-and-out inside the game’s final two minutes. The Giants took over at their own 29-yard line with 57 seconds remaining, trailing by seven and with no timeouts remaining.

McCarthy evaded pressure in the backfield on the first play of the drive, finding a wide-open gap in the Saints defense for a 28-yard scramble, then he hit Likely for a gain of 23 on the next play to push the ball inside the Saints’ 25.

But the Giants rally fizzled when McCarthy’s fourth-and-1 pass with 12 seconds remaining sailed high and out of bounds over Likely’s outstretched hands.

“Somewhere in there, the foot got off the gas pedal and the Giants came back in it,” defensive end Cam Jordan said. “I mean, kudos to them for being a tenacious team. We got to be able to snuff that fire.”

New Orleans was scheduled to take an overnight flight to Paris, where it will take on the Pittsburgh Steelers next week.