David Norris struck twice and Ronan Murray marked his debut with a goal as Town booked a place in the second round of the Carling Cup.
Blues needed extra-time to progress at St James Park and had to come from behind to win for the second time in four days, Murray cancelling out Ryan Harley's opener for the Grecians.
Norris put Town ahead on 73 minutes but Harley took the game into the added half hour with a superb finish from the edge of the box.
Former Blue Marcus Stewart could have won it for the home side before the 90 minutes but it was Norris, a cult hero up the road at Plymouth for many years before his move to Portman Road, who provided the vital third goal in the first-half of extra-time.
Roy Keane, celebrating his 39th birthday, made eight changes to the side that beat Middlesbrough at the weekend, with Troy Brown handed his first start for Blues.
Shane O'Connor returned to the side after missing the trip to the Riverside with a virus while Colin Healy made his first appearance in a Town shirt since the Boxing Day defeat at Crystal Palace.
Exeter included Stewart and fellow former Blue Richard Logan on their bench.
Stead could and perhaps should have followed up his goal at Boro on Saturday with the ball at his feet 12 yards out on six minutes but he failed to hit the target.
Exeter matched Town throughout the first-half and Matt Taylor wasted a great chance to put his side ahead on 25 minutes, heading over from Harley's corner.
Healy sliced wide after Grant Leadbitter's excellent free-kick eventually fell into his path, while at the other end Daniel Nardiello pulled his shot wide after a clever turn.
An opening half of few chances ended with Harley having a shot blocked and Reggie Lambe firing well wide.
Town made two changes inside the first seven minutes of the second-half, Lambe and Jack Ainsley making way for Murray and Jaime Peters.
It was the home side who made the breakthrough though, Harley beating Brian Murphy with an excellent finish on 53 minutes.
Town rallied, Stead and Norris were both denied then Murray forced Paul Jones into a save before Blues levelled and it was the&nbsp;Republic of Ireland U19 international&nbsp;who did the trick, beating Jones with a header from O'Connor's cross just past the hour.
Exeter made their first change with James Dunne on for David Noble. Tommy Smith, also a second-half substitute, on for the injured Brown, showed good awareness to clear the danger from Nardiello but it was Blues who looked the more likely to edge in front, Stead linking up well with Norris but just failing to find the net.
Blues got the goal the pressure threatened though on 73 minutes, the Plymouth midfielder beating Jones after Murray had set him up.
Exeter made a double change on 78 minutes with Logan and 37-year-old Stewart, a player coach at St James Park, introduced into the action, John O'Flynn and Richard Duffy making way.
The Grecians levelled the scores in spectacular style 10 minutes from time, Harley firing in the goal of the night with a terrific right foot strike into the roof of the net and it took a brilliant tackle from Smith to deny Stewart as he threatened a fairytale winner for the home side soon after.
Murray went down on the edge of the area on 90 minutes with Town looking for a penalty but referee Rob Shoebridge awarded a free-kick after consulting his assistant, Leadbitter's set-piece deflected wide.
Norris picked up the only booking of the game on 97 minutes for a foul on Dunne but the midfielder's next involvement was crucial, grabbing the vital third goal after Stead had twice&nbsp;been denied following an excellent Luke Hyam pass.
Murphy saved well from Logan, the Republic of Ireland U21 international called into action again to save from Nardiello as the Grecians searched for an equaliser.
Town finished with O'Connor and Tom Eastman suffering with cramp but a second win of the campaign and a place in the next round.
How they lined-up
Exeter: P.Jones, B.Jones, Taylor, Duffy (Stewart 78), Tully, Sercombe, Noble (Dunne 64), Harley, Golbourne, Nardiello, O'Flynn (Logan 78). Subs not used: Thomson, Krysiak, Cureton, Edwards.
Town: Murphy, Ainsley (Peters 52), Eastman, Brown (Smith 62), O'Connor, Hyam, Lambe (Murray 52), Leadbitter, Norris, Healy, Stead. Subs not used: Lee-Barrett, Carson, Griffiths.