Tottenham exited its second competition in four days after losing 2-0 at Crystal Palace in the fourth round of the FA Cup on Sunday.
Palace took a ninth-minute lead when Jeff Schlupp's shot was saved by Paulo Gazzaniga but the Spurs goalkeeper could only push the ball into the path of Connor Wickham and he scored off his thigh.
The hosts extended their advantage after Kyle Walker-Peters handled in the penalty area with Wickham lurking. Former Tottenham winger Andros Townsend slammed the resulting spot kick straight down the middle in the 34th.
Tottenham got a penalty of its own 10 minutes later when Patrick Van Aanholt brought down Juan Foyth, but Kieran Trippier fired well wide to ensure the Eagles took a 2-0 lead into the break, and then survived a second half dominated by Tottenham.
Chelsea beat Tottenham in a penalty shootout on Thursday in the English League Cup semifinals after a 2-1 victory for Chelsea at Stamford Bridge left the two sides level 2-2 on aggregate.
Defending FA Cup champion Chelsea hosts Sheffield Wednesday later in the other fourth-round game on Sunday.
The fifth-round draw takes place Monday.
