Jack Dyer, the Sky Bet League Two club's third penalty taker in the shoot-out, saw his spot-kick saved by David Stockdale.
That miss came immediately after Bryan Ruiz had a lucky escape - his penalty striking the post but ricocheting into the back of the net off unfortunate Albion goalkeeper Dean Lyness.
John Arne Riise belted home the final penalty to send Martin Jol's Premier League team through - but not after a scare.
Fulham broke Burton down in the 36th minute through Adel Taarabt.
Dyer sent the tie into extra time with an equaliser after 85 minutes, and Burton went ahead in the 102nd minute through substitute Michael Symes.
It looked like Gary Rowett's side would mark their first ever League Cup tie at the Pirelli Stadium with a famous victory but Fulham substitute Hugo Rodallega levelled with three minutes remaining to send it to penalties.
Fulham enjoyed the majority of the possession as Taarabt orchestrated the play. But Jol's team struggled to get much change out of their hosts.
The substantial figure of Rene Howe caused Fulham's centre-back pairing of Hangeland and Aaron Hughes a few problems in the first half.
The striker showed a good touch to spin beyond Hangeland just inside the box but shot harmlessly wide.
He then displayed strength and tenacity in retaining possession before forcing Stockdale to smother a low shot down by the post.
Those efforts came either side of Taarabt's goal, which followed some patient build-up play from the Premier League side.
Taarabt was involved numerous times before Scott Parker picked him out again on the left side of the box.
The Morocco international, who began limping midway through the first half, cut inside Phil Edwards and arrowed a right-footed shot beyond Lyness and into the far corner of the net.
The knock that had been hampering Taarabt was enough to prevent him emerging for the second half and he was replaced by Alexander Kacaniklic at half-time.
Jol was becoming increasing lively in his technical area as Fulham failed to put the tie to bed.
A defensive error let in Bent as the midway point of the second half approached but his half-volley was well blocked by Lyness who came rushing towards the edge of his area to meet the striker.
Soon afterwards a terrific clearance from Edwards denied Kacaniklic a simple tap in at the near post.
Burton's best spell of the match followed. Some sustained pressure resulted in the ball breaking to Dyer inside the area on the right-hand side and he fired a low shot narrowly wide of the far post.
The midfielder was finding his range, as with five minutes remaining Chris Hussey's left-wing cross found Dyer in a similar position and this time his right-footed shot found the bottom corner of the net.
Some inside the Pirelli Stadium thought Gary Rowett's team had won it when Michael Symes had the ball in the net again but the substitute had blatantly pushed it in with his hand and rightly received a yellow card for unsporting behaviour.
Burton's goal led a charmed life as extra time began.
Lyness tipped away a Bent volley and pushed another shot wide from substitute Pajtim Kasami before Sascha Riether's effort from the centre of the penalty area struck the base of the post.
So it was firmly against the run of play when Symes headed the Brewers in front, out-jumping Riise at the far post to nod home Hussey's centre.
Only a superb one-handed save to flick Jimmy Phillips' long-range effort over the crossbar prevented Fulham falling two goals behind.
It looked like Burton would hang on for a famous win but a scramble inside the home box after a Parker shot came back off the bar ended with Rodallega riffling a right-footed shot into the roof of the net.
Fulham would then edge through via the drama of penalties.