to come from behind and win 2-1 against Tottenham Hotspur, who missed a late penalty.
A strong Tottenham side went in front through Gareth Bale in the 66th minute, but a Jan Vertonghen own goal in the 84th minute levelled matters.
Norwich, who are struggling in the Barclays Premier League, completed the turnaround three minutes later when Simeon Jackson followed up to tap home after fellow substitute Grant Holt's header was parried by Hugo Lloris, Tottenham's French international goalkeeper.
Clint Dempsey had the chance to make it 2-2 from the penalty spot a minute later, but his low effort was kept out by Mark Bunn in the Norwich goal.
Brendan Rodgers' Liverpool side, the cup holders, went down to a 3-1 defeat at Anfield against his former club Swansea City.
Rodgers paid the price for fielding a weakened team as Chico headed Michael Laudrup's side in front just after the half-hour mark.
Nathan Dyer doubled Swansea's advantage 18 minutes from time and, even though substitute Luis Suarez managed to pull a goal back four minutes later when he headed home a Steven Gerrard free-kick, Jonathan de Guzman killed off any hopes of a recovery by scoring in time added on for stoppages.