Daniel Sturridge and Adam Lallana scored early in the second half as Liverpool fought back from conceding a 15th-minute goal to Fraizer Campbell at Selhurst Park.
West Bromwich Albion thrashed 10-man West Ham 4-0 in another all-Premier League matchup in the last 16, while second-tier Blackburn capitalized on the sending-off of Stoke's American defender, Geoff Cameron, to rally to a surprise 4-1 win. Reading won 2-1 at Derby in a match between two second-tier sides.
Titleholder Arsenal is at home to Middlesbrough on Sunday and another of the competition's favourites, Manchester United, visits third-tier Preston on Monday. Premier League giants Chelsea and Manchester City were knocked out in the fourth round.
If Liverpool reaches the final on May 30, it will be Gerrard's last game in the red jersey before he ends his career at Anfield and moves to the Los Angeles Galaxy in Major League Soccer.
The dream of raising one final trophy for his long-time club is still alive for the former England captain, who missed the Palace game with a hamstring injury.
Campbell put Palace ahead when he followed up to score after strike partner Dwight Gayle had a header parried by Liverpool goalkeeper Simon Mignolet.
Sturridge, starting for the second straight game on his return from five months out injured, met Jordan Henderson's pass with a brilliant volley to equalize in the 49th. Lallana grabbed the winner from close range in the 58th after Mario Balotelli's powerful free kick was blocked.
Nigeria striker Brown Ideye maintained his resurgent form at West Brom by scoring in either half in the convincing win over West Ham.
Ideye almost left West Brom in January, six months into a difficult first season, but stayed at the Hawthorns and has now netted for three straight games under recently hired manager Tony Pulis.
He put West Brom ahead in the 20th by turning in Craig Dawson's centre, and made it 3-0 with a 57th-minute header from Stephane Sessegnon's cross.
James Morrison scored the second in the 42nd and Saido Berahino completed the rout in the 71st, just after Morgan Amalfitano was sent off for West Ham for picking up a second yellow card for pushing West Brom midfielder Chris Brunt in the face.
Stoke went ahead at Ewood Park through Peter Crouch after 10 minutes, but conceded an equalizer to Josh King before Cameron got a straight red card just before halftime for fouling King in the area.
Rudy Gestede converted the penalty and King completed a hat trick with two goals early in the second half.
Derby was also reduced to 10 men for the second half at home to Reading when left back Stephen Warnock earned a second yellow card in the 39th.
Hal Robson-Kanu's goal for Reading was cancelled out by Darren Bent and former Everton and Middlesbrough striker Ayegbeni Yakubu — signed as a free agent during the winter transfer window — grabbed the winner with a left-foot shot in the 82nd.