Luis Suarez put Liverpool ahead in the 23rd minute, but Stoke hit back just three minutes later through a Peter Crouch header.
Stewart Downing came through when it mattered for the Merseyside club, and his goal - his first since making the switch from Aston Villa in the off-season - just before the hour mark put Liverpool into the last four.
They now head to Wembley in the last four of the competition, where they will meet either Everton or Sunderland.
Fernando Torres finally found his scoring touch as Chelsea comfortably beat Leicester City 5-2 to progress to the semi-finals of the FA Cup.
Gary Cahill and Salomon Kalou gave the Blues a half-time lead before Torres netted either side of a Jermaine Beckford strike.
A late wonder goal from Ben Marshall and a Raul Meireles strike gave the game an impressive scoreline.
The win sees Chelsea through to the semi-finals where they will be making their 10th trip to the new Wembley since its 2007 reopening.
Everton and Sunderland produced a scintillating 1-1 draw in their FA Cup quarter-final at Goodison Park on Saturday.
Phil Bardsley had opened the scoring for the visitors before Tim Cahill's opportunistic strike restored parity for the Toffees.
The two teams will now have meet again at the Stadium of Light to decide who will appear at Wembley.
The match between Tottenham Hotspur against Bolton Wanderers was abandoned after 41 minutes after Bolton's Fabrice Muamba suffered a cardiac arrest on the pitch. After attempts to resuscitate him, he was taken to the heart attack centre of London Chest Hospital, where he is said to be "critically ill" and in intensive care.
