Carlisle showed strength of character and determination to claim their play-off spot as they twice came from behind before putting themselves in the driving seat in a game which was thrilling, entertaining and fantastic to watch.
Exeter passed the ball well from the kick off but United bided their time and also had good spells of possession when they got themselves into the game.
Chances were few and far between as things stayed tight but a surging run from Wheeler brought a cross as he made it to the edge of the box. Gillespie watched it and pulled it into his chest.
Adams pulled wide to do the same at the other end but Brown was back to hook it clear before Ibehre could get there.
Disaster struck for the Cumbrians with just ten minutes on the clock when Wheeler and Reid beat the offside trap for a ball over the top. Wheeler clipped his pass through for his strike partner to chase and Reid made no mistake with a measured finish into the bottom corner.
United looked to respond and two deliveries from Adams caused problems as Ibehre and Proctor challenged. The ball wouldn't quite drop and Pym came away with it on both occasions.
Exeter found a way through the back four again as they got forward with Reid right at the heart of it once more. He brought Watkins into play and Gillespie had to go full length to beat it away. Wheeler was on to the rebound but Liddle stepped across to clear it to safety.
United were unlucky midway through the half when excellent control from Proctor took him past his man and into the area. It was a difficult angle but the keeper hit the panic button as the shot deflected off a defender and into the air. It dropped onto the top of the bar and fell behind for a corner.
The delivery from Adams from that set piece found Tom Miller on the move but he couldn't keep his header down and it flew into the stands.
Carlisle dominated most of the possession but they were almost caught on the break when Reid carried the ball out of his own half as a set piece broke down. Gillespie moved to cover his angles and he made a fine save when a driven shot came his way.
United were back on terms on the half hour when an Adams cross fell into the box. The referee looked at his assistant - who had spotted a pull on Proctor as he battled to gain space - and had no hesitation in pointing to the spot. Danny Grainger put the ball down and left the stopper with no chance as he crunched it into the roof of the net.
Reid twice had shots blocked as the Reds tried to regain the upper hand but United went close once more when Miller beat the keeper to an Adams corner. It rose high and was headed off the line by an attentive David Wheeler.
Carlisle were dealt a second blow on the stroke of half time when a regulation corner was flicked on by Joyce as he tried to get it away. Jordan Moor-Taylor was all alone and he made no mistake as he put it into the gaping net from just four yards out.
The second half took a while to get going but Exeter had to defend a corner from Adams and a long throw from Miller in quick succession. They did that well with Brown the man to make the clearances every time it came into the mix.
A third corner from Adams wasn't so easily dealt with as Miller arrived late to get a firm connection. The keeper was beaten but it was hooked off the line as the Carlisle contingent claimed for handball.
Cumbrian tails were up and a Proctor shot was blocked at source as he burst into the area. Substitute James Bailey wasn't too far away as he cut in from the left wing to have a go. His shot curled through bodies and just past the post.
It was all about United and Adams was next to have a pop as he let go with a volley when it dropped on his boot. The block brought Grainger into play and Devitt was put into a dangerous area via another through ball from Adams. It took the slightest of touches and Ibehre was frustrated to see it skim past his touch.
The pressure stayed on and Raynes jumped high to meet the latest corner from Adams. Pym saw it late but he did enough to claw it away.
A floated cross from Bailey was next to test the Grecian central defence but Joyce had to stretch and he couldn't keep it in play. Shaun Miller had a shot blocked as wave after wave of blue attacks continued and the pressure was finally made to pay when Proctor was upended inside the box.
Danny Grainger kept his composure with a repeat of his first half spot kick to bulge the roof of the net.
And the game was turned on its head just three minutes later when Tom Miller launched a long throw straight on to Proctor's head. He had to make room for himself but it did the trick as his back header fell over the line.
United kept going forward and a cut back to Shaun Miller from Adams had him interested. His shot had venom but it hit a thigh and ballooned high.
Carlisle had two late set pieces to contend with, but Tom Miller got to the first, and Raynes the second, and that was that as they saw the game out through added time to deservedly keep the season well and truly alive.
Exeter City - Reid (10), Moore-Taylor (45)
Carlisle United - Grainger (30 pen, 71 pen), Proctor (74)
Carlisle United - Grainger (89)
Exeter City - Pym, James (Tillson 70), Wheeler, Harley (Croll 70), Watkins, Moore-Taylor, Taylor (Grant 80), Ampadu, Sweeney, Reid, Brown. Subs - Olejnik, Holmes, McAlinden, Stacey.
Carlisle United - Gillespie, Liddle, Raynes, Gillesphey (Bailey 46), T Miller, Devitt, Joyce (O'Sullivan 70), Grainger, Adams, Proctor, Ibehre (S Miller 66). Subs - Crocombe, Tomlinson, Lambe, Waring.
Referee - Mr Stephen Martin
Attendance - 6,774 (735 away fan)