Marco Silva's side trailed to Cauley Woodrow's fourth-minute rocket against his former club.
The Cottagers, who have never won the League Cup, equalised when Josh King scored his first goal in senior football in the 48th minute.
Wycombe held out as Fulham put them under heavy pressure, with Anders Hagelskjaer clearing Kevin's strike off the line in the final seconds.
In the shoot-out, Fulham twice missed the chance to win it as Ryan Sessegnon and Jonah Kusi-Asare were denied by Wycombe keeper Will Norris.
But the hosts could not take advantage and, after Donnell McNeilly's kick was saved, Issa Diop thumped in the winner.
Fulham's win ended a run of four successive defeats that had sent them crashing down to 17th place in the Premier League.
Brentford joined their west London neighbours in the last eight with a 5-0 demolition of Grimsby that ended the giant-killers' memorable run in the competition.
The fourth-tier Mariners stunned Manchester United in the second round and knocked out Championship side Sheffield Wednesday in the third round.
But Grimsby were quickly subdued by two goals in five first-half minutes as the Bees took control before half-time despite making nine changes from the team that shocked Premier League champions Liverpool 3-2 on Saturday.
Mathias Jensen opened the scoring from 20 yards, Keane Lewis-Potter headed home a second, and Reiss Nelson effectively put the tie to bed before the break at Blundell Park.
Fabio Carvalho added Brentford's fourth goal from the penalty spot in the second half, and substitute Nathan Collins headed in the fifth to complete the rout.
The victory kept Keith Andrews' side in the hunt to reach their first-ever major cup final.
Will Fish's volley fired third-tier Cardiff into the quarter-finals as Championship side Wrexham were beaten in the all-Welsh clash at the Racecourse Ground.
Wrexham substitute Kieffer Moore equalised against his former club in the 52nd minute after Yousef Salech had given Cardiff the lead in the 13th minute.
But Cardiff were not to be denied as Fish's 71st-minute finish secured the bragging rights in the clubs' first meeting in 21 years.
The Bluebirds are in the quarter-finals for the first time since they reached the 2011-12 final.