Liverpool squandered a two-goal lead but Daniel Sturridge and Jordan Henderson ensured a 4-2 victory in extra-time.
The Hammers finally equalised in the 48th minute when Mark Noble played a quick free-kick that caught the hosts off guard and Nolan fired a low strike past Mark Schwarzer. But Fulham were ahead again within 60 seconds. Duff burst down the left flank and sent in a cross that Colombian striker Rodallega reached ahead of West Ham's defenders to bundle home.
Fulham gained their first away win of the season on Saturday with a 2-1 win over Wigan to extend their unbeaten streak against their opponents to 12 games.