The Reds were without Philippe Coutinho (thigh) and Mohamed Salah (groin) at Turf Moor on Monday, but Sadio Mane's stunning opener just after the hour had them in front.
Johann Berg Gudmundsson equalised for the Clarets in the 87th minute. However, Klavan stole in at the back post deep into stoppage time to head home from point-blank range and leave Klopp satisfied with a second win in three days following Saturday's defeat of Leicester City.
"It was a very difficult game, these fixtures are really difficult to take," Klopp said. "We did brilliant, the weather made it so hard, windy, rainy.
"Burnley are dong well, we denied them to do what they usually like to do. It took a while to score and then the next two goals were very similar.
"It was a deserved three points and if people say [otherwise], I don't care.
Happy New Year, Reds. pic.twitter.com/ZBjLHrXTnL— Liverpool FC (@LFC) January 1, 2018
"Six points in two days - I've never had that before. A fantastic finish to the game. This is one of the toughest places to play all season.
"I like rotation, I love the quality of my squad and I am really pleased. It was not fantastic football but it was a fantastic attitude.
"Today it wasn't possible to play brilliantly but it was an open game, we had a massive setback with the equaliser but we changed it again - we haven't done that too often this year.
"Adam Lallana was great, after four or five months out, performing like this. It's not easy at this time, and the whole team applauded Sadio Mane when he came in because we needed that goal. He is not playing at his most fluent but a great goal."