The hosts, who honoured club president Sir Jack Hayward before kickoff following his death earlier today, were slow to get going and once again found themselves on the back foot when McCormack and Sean Kavanagh put efforts over the top.
Fulham grabbed the lead shortly before the half-hour mark, however, with Woodrow latching on to a cross and nodding an effort past the keeper.
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The visitors should have doubled their lead 10 minutes before half time as Moussa Dembele raced through on goal before rounding the keeper, only to miss his resulting shot from five yards.
The hosts made two changes at the break and came within inches of an equaliser 20 minutes after the restart when Matt Doherty's header hit the post before the rebound was cleared off the line.
Fulham had the woodwork to thank again in the 69th minute as Nouha Dicko's strength saw him break into a shooting position, from which he crashed a powerful effort against the crossbar.
Wolves' increased pressure finally told shortly afterwards, however, as Dominic Iorfa did brilliantly down the flank before cutting the ball back for Edwards to slide home.
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The lead was gone as quickly as it came about, however, as Fulham restored parity almost immediately through Woodrow's second of the night, this time tapping home Dembele's shot.
Both sides looked to push for a winner in the closing stages of normal time, and Edwards almost provided it with a curling effort that flew inches wide of the post.
Neither side could avoid extra time in the snowy Molineux conditions, and it was the hosts who started the additional period brighter as Bettinelli denied both McDonald and Iorfa with fine saves.
Wolves were forced to wait until the 109th minute to get their third goal, however, as Batth brought the ball forward from midfield before crossing for Edwards to double his personal tally at the near post.
Fulham refused to give in, however, and added another late twist to the tale in stoppage time of extra time as McCormack's free kick hit a hand in the area, leading the referee to point to the spot.
McCormack stepped up to take the kick himself and made no mistake, sending Ikeme the wrong way to force a shootout.
Fulham then posted a perfect record from the spot, scoring all five while Doherty missed for the hosts as the Cottagers ran out 5-3 victors in the shootout.
Kit Symons's side will now face Sunderland in the fourth round later this month.