A crowd of 11,416 watched on at Molineux as neither side managed to find the net in 120 lifeless minutes, despite the Barclays Premier League outfit dominating most of the possession.
Coca-Cola League One Swindon deserved credit for their brave effort but, in truth, Wolves fell far short of expectations.
Robins wide man Anthony McNamee arguably came the closest the breaking the deadlock when his first-half strike bounced off the top of the crossbar, while team-mate Mark Marshall and Wolves striker Sam Vokes also saw tame efforts strike the woodwork.
And almost predictably it needed spot-kicks to settle matters, with the first 11 players all finding the net before Robins captain Gordon Greer saw his effort come back off a post.
The victory brought an end to Wolves' recent dismal record in the tournament having gone out to the likes of Chesterfield, Morecambe and Rotherham in recent years.
Mick McCarthy turned to his fringe first-team players for the encounter, naming an entirely new XI from the side that started Saturday's 1-0 defeat to Manchester City.
Yet it remained a strong line-up with Marcus Hahnemann, Ronald Zubar, Andrew Surman and Kevin Doyle all handed competitive debuts.
Youngster Nathaniel Mendez-Laing, 17, also making his first start for the club.
Swindon boss Danny Wilson, meanwhile, made three changes from Saturday's 2-2 draw at Oldham.
Full-backs Scott Cuthbert and Callum Kennedy and midfielder Craig Easton coming in for Sean Morrison, Lescinel Jean-Francois and Michael Timlin.
Doyle had a great chance to register his first goal in a gold and black shirt after just three minutes.
Sam Vokes latched on to Hahnemann's long kick, laid the ball off to Doyle, who cleverly turned his man before firing narrowly wide from the edge of the penalty area.
Yet, in front of a half-full Molineux, the game quickly fell into a lull.
It needed a curling effort from young academy prospect Mendez-Laing to bring the clash back to life, a strike that was heading for the top corner until David Lucas palmed it over the bar.
After a flurry of activity, excitement soon slowed right down again, but the lively Mendez-Laing brought the home crowd to life with a jinking run and tame shot that was easily held by Lucas.
But Swindon soon proved their attacking intent in the 37th minute courtesy of a rare foray into the final third.
McNamee picked up the ball in plenty of space down the left-hand side and curled a looping effort that had Hahnemann beaten but bounced off the top of the bar.
Billy Paynter then silenced the home crowd further a minute before the interval when he prodded the ball into the net after Tope Obadeyi had seen his shot come back off a post.
But the assistant referee's flag was soon raised to deny the Robins what would have been a dream end to the first half.
Despite Swindon's on-loan Bolton forward Obadeyi blasting over, the second half soon fell back into the same routine as the first.
Mark Marshall was introduced for Swindon while both Andy Keogh, the only player to have found the net for Wolves this term, and Matt Jarvis also came on.
The changes seemed to add an extra attacking impetus to Wolves' play and Neill Collins headed a good chance over from a corner soon after.
There was a slight glimmer of hope as substitute Mark Marshall made space for himself in Wolves' penalty area before harmlessly hitting a post as he looked to cross, while at the other end Vokes also saw a header clip the woodwork although Lucas had the ball well covered.
Keogh, by far Wolves' likeliest chance of breaking the deadlock, lobbed inches over having been put clean through, much to the anguish of the home support.
And a goal never came as a goalless 90 minutes was brought to a close, signalling an extra 30 minutes.
Dave Edwards tested the reliable Lucas with a stinging drive as extra time got under way and both sides huffed and puffed in the second period but, almost predictably, neither managed to find the net to send the tie to penalties.
The first 11 spot-kicks were all scored as both sets of players found the net at will, before Greer blasted against a post to put Wolves into Saturday's draw for the third round.