Palace, which has lost both of its games back in the topflight, was beaten 2-1 in Bristol.
Sunderland avoided a similar upset by fighting back from 2-0 down at home to another League One side, MK Dons, to win 4-2.
U.S. striker Jozy Altidore started the comeback by scoring his first goal for the club before Connor Wickham netted twice and Adam Johnson drilled in the fourth.
Liverpool was 2-0 up against Notts County, but the League One side scored twice to force the game into extra time when the hosts re-asserted their authority to win 4-2.
Liverpool opened the scoring in the fourth minute through a fine solo effort from Raheem Sterling. Daniel Sturridge then made the most of a defence-splitting pass from Steven Gerrard to put the hosts 2-0 up in the 29th.
But County pulled one back from Yoann Arquin's header in the 62nd, and substitute Adam Coombes slotted in a late equalizer.
But in stoppage time Liverpool recovered its superiority. Sturridge stroked in his second goal of the night and fourth of the season, before substitute Jordan Henderson netted Liverpool's fourth.
"I think we got complacent and they got some belief," Sturridge told British broadcaster Sky Sports. "They started playing better football, but our fitness from pre-season showed at the end."
Martin Jol's Fulham edged past fourth tier side Burton in a penalty shootout after a dramatic 2-2 draw.
Fulham keeper David Stockdale saved the crucial spot-kick from Phil Edwards to send the Premier League club through after they fought back from behind in extra-time.
Adel Taarabt scored his first goal for the Cottagers to put the visitors ahead in the 36th, but Jack Dyer forced extra time with an 85th minute equalizer.
Michael Symes put the League Two side ahead in the first period of extra-time, but Hugo Rodallega ensured penalties would settle the second round match when he levelled with three minutes to go.
At Leyton Orient, Robbie Brady's 106th-minute goal sent Premier League newcomer Hull through with a 1-0 victory over the League One leaders.
Across east London, Ravel Morrison marked his full West Ham debut with his first goal for the club to clinch a 2-1 victory over Cheltenham of the fourth tier at Upton Park.
Sweden striker Johan Elmander marked his full Norwich debut with two goals in the Premier League side's 6-3 win over Bury.
Saido Berahino marked his first West Bromwich Albion start with a 36-minute hat-trick to clinch a 3-0 win over Newport.
Goals from Steven Davis and Jay Rodriguez sent Southampton on its way to a 5-1 victory over Barnsley.