Enner Valencia opened the scoring for the hosts six minutes into the second half of the Upton Park replay but Kevin Mirallas netted a stunning 82nd-minute free-kick to force extra-time despite Everton having Aiden McGeady sent off.
Romelu Lukaku scored against West Ham for the fifth time in as many games for Everton but United responded through Carlton Cole's 113th-minute equaliser.
That meant a lengthy bout of spot-kicks but, after preventing Stewart Downing from winning the tie with West Ham's fifth kick, Everton goalkeeper Joel Robles clattered the crossbar before opposite number Adrian secured a 9-8 win for Sam Allardyce’s team on penalties.
The London club will travel to League One high-flyers Bristol City, who sunk Doncaster Rovers 2-0 thanks to a brace from Jay Emmanuel-Thomas.
Championship duo Wolverhampton Wanderers and Fulham also settled their match on penalties after a 3-3 draw at Molineux,
Cauley Woodrow headed Fulham into a 27th-minute lead but the forward's second goal of the evening was an equaliser forcing an additional period as Dave Edwards and Rajiv van La Parra netted in quick succession for Wolves.
Edwards looked to have sealed a trip to Sunderland with his 109th-minute goal but Luke McCormack levelled with a last-minute penalty before Fulham proved superior from 12 yards - converting all five kicks as Matt Doherty missed Wolves' second.
Elsewhere, Sam Clucas scored twice in extra-time as Chesterfield beat Scunthorpe United 2-0, securing a short trip to Derby County and the League One team’s first progress to the fourth round in 18 years.
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