A minute’s silence was observed before the game for the former Wolves owner Sir Jack Hayward, who passed away in Fort Lauderdale in Florida earlier on Tuesday.
Cauley Woodrow gave the visitors the lead on 27 minutes when he leapt above the Wolves defence to head home a Konstantinos Stafylidis cross.
David Edwards half-time introduction lifted the home crowd and he drew his side level on 71 minutes before Rajiv van La Parra thought he’d won the game for Wolves just two minutes later.
However, Woodrow grabbed his and Fulham’s second of the evening to force extra time.
Snow started to settle in the additional 30 minutes and Edwards found the net again with 11 minutes remaining, but Fulham won a penalty in the final minute of extra time and Ross McCormack coolly slotted the spot-kick home.
In the penalty shootout, Fulham found the net with all of their penalties while Matt Doherty’s miss proved costly for Wolves.
The west Londoners will play Sunderland in the fourth round.
Wolves started slowly and Carl Ikeme parried an early sighter from McCormack before Liam McAlinden was denied by Marcus Bettinelli in a rare home raid.
Fulham were comfortably on top and rubber-stamped their authority with the opener after 25 minutes when Woodrow powered in a 12-yard header from Kostas Stafylidis’ left-wing cross.
The opener allowed the visitors to continue to dominate against the lethargic hosts who were unable to cope with Fulham’s sharp passing.
Wolves were stifled when they did have the ball, leaving Bettinelli a spectator in goal.
And the Cottagers should have doubled their lead eight minutes before the break when Moussa Dembele wriggled through but, off-balance and under pressure from Ikeme and Richard Stearman, he shot wide with the goal gaping.
Wolves threw on Edwards and Nouha Dicko for the ineffective James Henry and McAlinden to change the game at the break.
And a quiet second half burst into life when the hosts were inches away from an equaliser as Doherty smacked the post from Van La Parra’s corner, before his follow-up was hacked off the line.
Two minutes later Dicko shuddered the bar from the edge of the area as a previously static Wolves continued to threaten.
And they deservedly levelled with 19 minutes left when Edwards prodded in Dominic Iorfa’s cross from eight yards.
Less than three minutes later Van La Parra put Wolves ahead when he chipped over Bettinelli after beating the offside trap.
The lead lasted just a further three minutes, though, as Woodrow made it 2-2 when he slammed in from close range after Ikeme parried Dembele’s shot.
The sides failed to find another breakthrough until the 109th minute when Wolves went ahead again. Danny Batth’s barnstorming run saw him cross from the right and Edwards converted from close range.
But McCormack stroked in a spot-kick at the death after Lee Evans handled his free-kick to send the tie to penalties.
Doherty was the only player to miss and Hugo Rodallega scored to send Fulham through.