United had to come from behind twice to get a point which keeps them right in the hunt with a play-off race which continues to take unexpected twists and turns.
Early sparring brought little in the way of goal mouth action but an overlap from Holmes earned him a corner with six minutes on the clock. His delivery was good but Kennedy twice intervened before he eventually hooked it clear.
The Grecians were gifted a lead on 7 minutes when a long range shot from Stockley bounced awkwardly in front of Gillespie. It spooned up off his elbow as he dived low and he could do nothing as the spin on the bounce took it into the back of his net.
The game was then subject to a 65-minute delay as safety concerns over a fire alarm in the main stand meant that the players had to leave the pitch so that the supporters, press and officials could leave the seats for the relative safety of the playing surface.
Carlisle started well after the unscheduled break and a long throw from Grainger brought a flick from Eillis which caused real problems. Miller was an inch away as he jumped against Ribiero in a bid to get a touch.
Carlisle went close again when Hope broke through to chase a flick from Joyce. It sat up nicely and Olejnik had to watch closely as a lobbed volley threatened to creep in.
The Cumbrians conceded a free kick 22-yards out at the other end when Atkinson handled with Holmes on his tail, and it was Holmes who shaved the outside of the post as he bent a low shot round the wall with Gillespie unsighted.
The same player went close again when he beat the offside trap with an intelligent run. He twisted and turned but he couldn't keep his chipped shot on target.
Gillesphey did well to stop Wheeler from going one-on-one as Exeter started to send men forward, but United were back on terms when Charlie Wyke showed a real striking instinct when the hosts lost control of the ball deep inside their own half. The big forward used strength and pace to get into the box and his final touch left Olejnink with no chance as it ran unhindered into the bottom corner.
A misplaced back header from Miller was almost punished as Wheeler found himself with just Gillespie to beat. The United keeper closed the ground quickly and made a superb and vital save with his boot.
A run from Hope brought a shot into the stands as Carlisle looked to do damage on the break, and a dipping cross from Hope found Grainger on the move as he arrived late at the back post. His header had direction but Olejnik was there to grasp it.
Grainger sent a long range free kick over the bar as United kicked the second half off on the attack, but a set piece at the other end signaled Exeter's arrival into the half when Holmes bent it low. Stockley threw himself at it but he couldn't keep the header down.
The Cumbrians thought they'd taken the lead on 54 minutes when Ellis won the flick from a Miller throw. Kennedy stretched to meet it but Olejnik got enough of a touch to send it onto the underside of the bar. A scramble followed as Hope and Wyke tried to make it count, but the ball was eventually cleared as blue shirts converged.
Exeter attacked in numbers and a rattled a cut-back from Wheeler was smashed through legs by Harley, and it needed a block from Atkinson as it angled at the target.
A promising run from Gilliead was brought to a halt as his pass to Hope was waved offside, and Stockley controlled acrobatically only to volley wide as the action swung from end to end. Stockley then lifted the ball into the away fans as he came under pressure from Gillespie and Gillesphey when the game started to open up.
Carlisle passed their way forward, as they started to exert some pressure, with Kennedy and Gilliead right at the heart of it but the final cross from Kennedy was easily gathered by Olejnik. Watkins dragged one wide at the other end when he was first to a ball over the top, but Gillespie had it under control as it flashed across his chest.
A difficult volley for Joyce arrowed over the bar as United continued to ask questions, but they were undone for a second time on the day when Holmes dropped a shoulder and delivered a cross. Wheeler ran off his man and finished it off with a well taken volley.
The lead was short-lived and a determined Carlisle side kept their play-off hopes alive with a decisive attack. Joyce fooled everyone with a shot from the outside of the boot which bounced down off the bar, and Hallam Hope remained alert to get there first with a header which earned his side their deserved share of the spoils.
Carlisle won a late, late corner through Gilliead and they went close as the Grainger delivery somehow stayed out as it hit a defender's leg and bobbled away to safety.
Exeter - Stockley (7), Wheeler (79)
United - Wyke (28), Hope (85)
Exeter - Woodman (47), Oakley (53), Watkins (61)
United - Grainger (80), Gillesphey 82)
Exeter City - Olejnik, Woodman, Ribeiro, Harley, Oakley, Holmes, Moore-Taylor, Wheeler, Watkins, Stockley, Brown. Subs - Noble, McAllister, Morrison, Tillson, Pym, Grant, Taylor.
Carlisle United - Gillespie, Miller, Atkinson (Raynes 74), Ellis, Gillesphey (Asamoah 84), Hope, Joyce, Kennedy, Grainger, Wyke, Rigg (Gilliead 60). Subs - Hanford, Hery, Brough, McQueen.
Referee - Mr Lee Swabey
Attendance - 3,375 (172 away fans)