A tight and tense tie needed a penalty shoot-out to take United through to a lucrative fourth round meeting with Everton at the end of the month.
The game started at a frantic pace but Gilliead imposed order when he robbed possession and went on a run in midfield. He exchanged passes with Hope and a lay-off for Sweeney brought a crisp shot which angled over the bar.
United kept pushing forward and a long throw brought two volleys from Sweeney which ricocheted back off Dolan. Raynes stepped in but his scuffed shot was dragged well wide.
A slight opening for Zoko at the other end brought a long range thunderbolt which flew off into the night sky, and a lobbed ball over the top presented Compton with a difficult chance as it skipped up off the surface. It was a tight angle and he couldn't keep it down.
Compton was wasteful again in the next attack when he cut back from wide to let one fly. It had power but it lacked direction and purpose.
Clear-cut chances were at a premium and Roberts wasted a decent chance when a half clearance dropped at his feet. He could have taken a touch but he opted for a first time punt which spun harmlessly away.
Carlisle created an excellent chance from a Grainger corner at the other end as Yeovil struggled to clear the inswinging delivery. It span high but Sweeney could only direct his header right back at Krysiak.
Disaster struck for Carlisle just after the half hour when a lightning quick break caught them short in numbers at the back. A dummy from a slick Jeffers pass left Compton with work to do on the edge of the box, but a drop of the shoulder and a whipped left footed shot left Gillespie with no chance as the ball bent into the corner of his net.
United had a run of set pieces at the end of the first half but the Glovers held their discipline and went into the break with their one-goal advantage.
Manager Keith Curle went for a double substitution after the break as Ibehre and Gillesphey replaced Gilliead and Grainger, and United had the first attack when Joyce swung the ball in from wide. Sweeney won the header but Krysiak was quickly off his line to claim it.
A deep cross from Walsh brought a glanced header from Zoko as the second half struggled to get going, but Gillespie had watched it all the way and he gathered it comfortably.
The Glovers were a whisker away from doubling their advantage, as they spread play from left to right, with the dangerous Jack Compton at the heart of the action. His rasping effort had the keeper beaten but it grazed the top of the bar.
Another Compton effort bobbled wide as the hosts pushed men forward, but Carlisle had half a chance when Gillesphey took charge of a 28-yard free kick. He got it over the wall but it bounced wide of the upright.
A run from Asamoah brought a shot which deflected off an outstretched boot and Krysiak was relieved when it skipped off the surface and went behind for a goal kick. The keeper then had to tip a looped header from Sweeney over as United tried to turn up the pressure levels, but Ellis had to block from Zoko at the other end when Yeovil hit back on the break.
The home side were left to wonder how they hadn't scored with 15 minutes left to play when a Compton corner flicked off Raynes. The ball flew across the face of goal but there was nobody there to apply the finishing touch.
United sent men forward in numbers and they were back on level terms when Yeovil failed to deal with an inswinging corner. The ball was twice hooked off the line but Sweeney kept his eye on it and showed excellent determination as he forced his way through to the head it into the top corner.
Carlisle started to look like a confident side and Sweeney smashed a great volley onto the crossbar as he burst through from deep. The rebound fell to Wyke and Krysiak had to dive long to parry his follow up away.
Yeovil captain Francois Zoko wasted a chance to win the tie from the spot in time added on when Gillespie felled him inside the box. He placed the ball but sent a very poor spot kick wide.
Yeovil defender Alex Lacey was shown a straight red card for a late challenge on Wyke in the first half of extra time, and United midfielder Luke Joyce joined him seven minutes later when he was given a second yellow for a sliding challenge on Jeffers.
The best Carlisle chance of a tight extra time showing fell to Ibehre when he nicked the ball to go one-on-one with Krysiak. The keeper spread himself and dived low to make a vital block at the striker’s feet.
A deflection from a Tozer shot was the closest Yeovil came as things remained tight, and Gillespie had to go full length to stop a lovely header from Sokolik from finding its way in. The big man went close again from the resulting corner, but it was all-square and penalties at the end of a tie which had given us just about everything.
Dickson - Left footed into the bottom corner - 1-0
Kennedy - Drilled into the left hand corner - 1-1
Jeffers - Sent Gillespie the wrong way - 2-1
Sweeney - Also sent the keeper the wrong way - 2-2
Tozer - Bottom right, rolled in - 3-2
Wyke - Straight down the middle - 3-3
Dawson - Another confident bottom corner strike - 4-3
Ibehre - No mistake, crisp and clean - 4-4
Dolan - Hits the base of the post [MISS] - 4-4
Ellis - Drilled it in - WON 5-4
Yeovil - Compton (31)
United - Sweeney (76)
Yeovil - Roberts (34), Walsh (63), Krysiak (79), Sokolik (114)
Sent off: Lacey (97)
United - Hope (37), Atkinson (42), Ibehre (79)
Sent off: Joyce (90 + 104)
Yeovil Town - Krysiak, Tozer, Dolan, Sokolik, Jeffers, Dickson, Compton (Dawson 89), Lacey, Roberts, Walsh (Smith 100), Zoko. Subs - Weale, Bird, Campbell, Gillett, Allen.
Carlisle United - Gillespie, Atkinson, Ellis, Raynes, Grainger (Gillesphey 46), Gilliead (Ibehre 46), Kennedy, Joyce, Sweeney, Wyke, Hope (Asamoah 65). Subs - Hanford, Miller, Hery, Dicker.
Referee - Mr Tim Robinson
Attendance - 4,114 (210 away fans)