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The Blues lost two players to first-half red cards - but defied the odds with a truly magnificent performance.
Carlisle led early on through Josh Williams before ref Jamie Robinson sent off both Miguel Azeez and Harvey Bunker inside the first half-hour.
United, though, showed remarkable resolve and went on to score twice more in the second half through Callum Whelan and Ryan Galvin.
Their goals were accompanied by some terrific defending and a surging atmosphere to the very end as the Blues defeated high-flying Harrogate for the first time in league football and moved up to ninth in the table.
The win was accompanied by memorable scenes in the stands as fans in the 6,225 crowd rallied behind Rob Elliot's team and their remarkably spirited victory.
Josh Williams celebrates Carlisle's opener (Image: Ben Holmes)
New signing Callum Camps started on the bench as Elliot made three changes to his XI.
Kenton Richardson made his first league start with David Ajiboye andAzeez also coming in.
They took the places of Tom Anderson, Regan Linney and Teddy Jenks, with Anderson ruled out with injury.
Harrogate boss Weaver made four changes as George Thomson, Morgan Williams, Ellis Taylor and Ciaran McGuckin replaced Michael Craig, Bailey Hobson, Will McGowan and Lewis Payne.
Weaver's side settled into their possession game early on but Carlisle pounced to take the lead on the ten-minute mark.
After Brad Nicholson lost possession, United quickly regained it as Whelan fed Williams on the left - and he cut in against his man and worked space to curl and impressive low shot into the far corner of the net.
Carlisle forced another chance soon afterwards as Azeez shot just wide when set up by Whelan's free-kick.
Miguel Azeez is sent off in the 14th minute (Image: Ben Holmes)
But then United were reduced to ten men with Azeez's dismissal.
Carlisle tried to play out from the back, Adam Desbois finding Feeney who turned it further down the left to Azeez, but the midfielder's touch inside left him stretching and as he slid into the challenge, Williams went down and the Harrogate bench was in uproar.
Ref Robinson pulled the red card from his back pocket, deeming it a dangerous challenge, and Carlisle were a man down.
Harrogate searched for a way back with Josh Hmami unable to direct a header past Desbois, before Richardson went into the book for a challenge - as did United No2 Louis Storey for his protests.
McGuckin twice went close for Harrogate - and then things took an even more controversial turn as Carlisle lost another player.
This time it was Bunker shown a straight red for his challenge on Hmami - as United protested that an offside decision should have gone in their favour beforehand.
The midfielder's raised foot in the challenge drew the decision from Robinson but Blues fans, coaches and players raged at the officials, feeling there had been minimal if any contact.
Nine-man United now went onto the defensive as they sought to regroup.
Kenton Richardson with a strong challenge (Image: Ben Holmes)
Brad Dolaghan shot over at the near post for Harrogate, before at the other end Morgan Feeney's header forced Luke Hutchinson into a save for the visitors.
After that, Elliot sacrificed Ajiboye and introduced Camps for his debut as Carlisle reorganised.
McGuckin failed to convert a far-post header for the visitors before Hmami and Williams also failed with attempts for Harrogate, Carlisle's nine men then cheered off by home fans at half-time as ref Robinson and his colleagues were loudly jeered.
And right at the start of the second half, the Blues stunned Harrogate with a second.
Williams did excellent work on the right to beat his man and when he cut the ball across the box, Whelan was the spare man to collect and slot home.
Brunton Park erupted, and Elliot's side then saw off a Taylor attempt before, remarkably, adding to their tally.
Galvin had only been on the pitch for five minutes after replacing Williams as part of a double change that also saw Jenks on for Whelan.
And just after the hour mark the defender was suddenly released on the break and found himself behind the Harrogate back line and going through on goal.
A defender got back to challenge as keeper Hutchinson also advanced - but the ball broke forward and Galvin was able to hurdle the Harrogate players and tap into an empty net.
Blues fans erupted once more at the unlikely scenes and United now had a three-goal lead to protect.
There were then heroics from captain Feeney as he cleared a Thomson finish off the line, before Desbois excelled to save from Dolaghan.
Elliot made further changes, sending on Stephen Wearne and Jenks, with Dolaghan again unable to beat Desbois at the near post before Jenks was denied a Carlisle penalty as he cut in and went down under a challenge.
United's second scorer Callum Whelan battles for the ball (Image: Ben Holmes)
Harrogate resumed their search for a way back but found Desbois in defiant form as he saved from Hmami, before Dolaghan saw a volley deflected over the bar.
Carlisle's resolve had Brunton Park in fine voice as the Blues moved ever closer to a memorable result, Desbois producing more outstanding goalkeeping to support the defiance of those in front of him.
Taylor skied a free header for Harrogate with two minutes of the 90 remaining, and United saw out four added minutes - with fans raising the roof again as the final whistle went, completing one of Brunton Park's most incredible nights for quite some time.
United: Desbois, Richardson (Mullarkey 68), Feeney, Nicholson, Duru, J Williams (Galvin 56), Bunker, Azeez, Whelan (Jenks 56), Ajiboye (Camps 40), Obiero (Wearne 68). Not used: Mair, Linney.
Goals: Williams 10, Whelan 46, Galvin 62.
Booked: Richardson, Storey (coach). Sent off: Azeez, Bunker.
Harrogate: Hutchinson, Baptiste, Dreyer, Cass, Bristow, Williams, Taylor, Thomson, Hmami, McGuckin (Muldoon 65), Dolaghan. Not used: Watson, Payne, Burrell, McCoulsky, Craig, Fox.
Booked: McGuckin, Baptiste, Williams, Dreyer.
Referee: Jamie Robinson.
Crowd: 6,225 (170 Harrogate fans).