Lacking the intensity of a regular league match there was little to recall in front of goal from the opening 45 minutes.
Olufela Olomola stung the hands of City’s keeper whilst Lee Martin went close at the other end.
Immediately after the break, Jordan Green had a one-on-one chance saved whilst Jayden Stockley fired just wide at the other end.
A livelier affair in the second half it remained goalless to earn both sides a share of the spoils before the Grecians won the penalty shootout to claim a second bonus point.
With no game for 10 days due to the forthcoming international break, Darren Way opted for a strong looking starting eleven with just four changes from the weekend’s win over Grimsby.
Saturday’s match-winner Omari Patrick came in for a home debut whilst Olomola, Jake Gray and Alefe Santos completed the changes.
Dropping out of the squad, Benin-bound Sessi D’Almeida earned a rest whilst Alex Fisher, Diallang Jaiyesimi and Alex Pattison took a seat amongst the substitutes.
Rounding off the changes to the matchday squad, academy defender Daniel Ojo earned a first call-up to take the number 26 squad number.
In a less ferocious edition of the usual West Country derby the opening quarter-of-an-hour came and went with little action of note.
Matt Taylor’s side enjoyed the better of early territory but couldn’t muster an attempt on goal, something the Glovers managed on 16 minutes.
In the game’s first chance, Olomola let fire with a venomous strike from the edge of the box which James Hamon could only push away.
Bringing the game to life, the Grecians responded with a good chance of their own a little under 10 minutes later.
Hiram Boateng’s neat through ball picked out the run of Martin who was thwarted by a smart Nathan Baxter stop when through on goal.
From the resulting corner, Archie Collins saw a well-hit effort fly just over the bar before chances dried up once again.
In an intriguing tactical battle neither goalkeeper was particularly threatened again until 10 minutes before the break.
With plenty of fluidity in the final third, Green and Olomola exchanged a neat one-two to work the ball into the box where an off-balance Green could only fire straight at Hamon.
As the break approached, Stockley had the ball in the net from a Martin cross but the linesman’s flag was up long before City’s skipper tapped in from close range.
The last action of a largely uneventful opening 45 minutes, both sides headed into the break with little to reflect on in front of goal.
With a change of tempo instantly recognisable after the restart, Town should have taken the lead just 90 seconds into the second half.
Yoann Arquin’s through ball saw Green clean through after Dara O’Shea’s slip up, only for Hamon to race from his line and thwart the diminutive attacker.
In a far more open affair, Stockley flashed a strike from the angle just wide for the visitors before the green and whites went close once again.
Arquin’s low ball from the right made it’s way through to Patrick on the left, with the winger’s goal-bound strike taking a vital deflection off of Jimmy Oates as the chances began to flow.
Keen to kick-start their cup campaign with a home victory, the regular starters of Alex Pattison and Diallang Jaiyesimi were introduced for Santos and Green but it was an Exeter sub who was next to go close.
Robbing Town of possession, Martin played the ball in for Chiedozie Ogbene who could only fire just over the bar when under pressure in the box.
Now into the final 10 minutes and with a penalty shootout looming Pattison almost snatched victory for the Glovers when his deflected strike fell inches wide, but an opening goal remained elusive as the tie headed penalties.
As with every shootout there had to be a villain, with Patrick blazing his spot kick over the bar to allow Jack Sparkes to claim the extra point.
Yeovil Town: Baxter; Mugabi, Sowunmi, Warren, Dickinson; Green (Jaiyesimi 65′), Santos (Pattison 52′), Gray, Patrick; Arquin, Olomola.
Subs not used: Phillips, Ojo, Rogers, Fisher, Henry.
Exeter City: Hamon; Oates, O’Shea, Croll, Woodman; Tilson, Boateng (Ogbene 56′); Martin (Abrahams 77′), Collins; Stockley (Sparkes 65′), Jay.
Subs not used: Dean, Sweeney, Law, Norman, Sparkes.