Nicolas Jackson scored the only goal of the game as the Blues picked up a much-needed victory following Sunday’s home defeat to Aston Villa in the Premier League.
Mauricio Pochettino has a far more realistic chance of winning this competition to deliver silverware during his first season in charge of the club and a pathway is opening up.
The Blues will host Championship side Blackburn Rovers in the fourth-round and success in the cup could also help them qualify for Europe, amid some differing League form.
Palmer played a key role on his first start for the club since a £40m move from Manchester City last month, operating as the No.10 and ultimately assisting Jackson.
Also able to operate out wide, the England U21 international insists he is willing to play anywhere under Pochettino.
“I enjoyed playing in that role tonight but to be totally honest, I’m comfortable in any of the attacking positions,” he said.
“Whether that’s as a No.10, out wide on the left or right or even as striker, I’m happy to do a job for the team.
“I like to get on the ball and that role allows me to do that but wherever the manager needs me, I’m happy to play.
“We’ll enjoy this win, especially after the last three results, but now it’s important we build on this ahead of Monday’s game.”