The home side enjoyed the better of the early exchanges, with their Premier League visitors struggling to find any rhythm going forward.
Neither keeper had been seriously tested by the halfway point of the first half, but soon after that David Martin had to get down to field a low 20-yard effort from Beram Kayal, who then tested Martin again with a header.
That proved to be a rare piece of goalmouth action in what was a disappointing first 45 minutes which ended goalless.
Early in the second half Albion keeper Mat Ryan kicked an attempted clearance against Shane Duffy, but recovered just in time to collect the loose ball before Lee Gregory could pounce.
Ten minutes into the second period Martin got down smartly to his right to keep out a right-footed effort from the busy Kayal, while at the other end Ryan stood firm to block a Wallace drive from a narrow angle.
Ryan was there again to thwart Gregory after he sprang the offside trap, as the game started to produce openings at both ends.
And one of those chances was taken 20 minutes from time as a Shaun Williams corner from the left was met by an unmarked Alex Pearce, who powered home a header from six yards out.
Eleven minutes from time it appeared to be game over as Wallace cut the ball back from the right side of the area and Aiden O'Brien swept his finish into the far corner of the net.
Late on, substitutes Solly March and Jurgen Locadia combined for the latter to flash a superb left-foot shot past Martin to halve the deficit.
And right at the death Brighton found an equaliser, as March won a free-kick 30 yards out, which he curled goalwards and Martin parried it into his own net, sending the game into extra-time.
In the first half of extra-time Ryan made a smart save to deny James Meredith, with Mahlon Romeo belting the follow-up high and wide when he should have done better.
However, it was the Premier League outfit enjoying the better of the play for the most part, although they were unable to carve out any clear openings, while Shane Ferguson was red-carded right at the end for a dreadful stamp on Duffy.
There was still time for Locadia to lash the ball home from a Martin Montoya cross, which was incorrectly ruled out for offside, before Albion prevailed 5-4 in a penalty shootout, with Jake Cooper missing the final spot-kick.