The result extends Liverpool's unbeaten run in all competitions to four following a draw with Tottenham Hotspur and wins over Leicester City and Chelsea.
After beating Derby, Klopp told Sky Sports News: "[I'm] happy of course. We are in the next round, which is the main thing. We were the better team, but we could have done better.
"The patience we had in the first half was nice, we could have passed a bit better but that's how it is. We created chances without giving them chances, which is very important. It was a good performance but that's what I expected for the game."
Next up for Liverpool is a league match against Hull City at Anfield.