Johann Berg Gudmundsson's stooping header in the 87th minute for the home side looked to have frustrated the visitors before Estonian defender Klavan bundled the ball in at the back post from a Dejan Lovren header to snatch all three points.
A stunning 61st minute strike from Sadio Mane, who blasted the ball home on the turn from the edge of the area, had put fourth-placed Liverpool ahead as they moved to 44 points.
It was a tightly contested game and Liverpool manager Juergen Klopp was delighted with his side's spirit.
"Did we deserve to win? I don't care. We got the three points," said the German. "It was a fantastic finish to the game. This is one of the toughest places to play all season.
"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."
Klopp made seven changes from the side that beat Leicester City 2-1 at Anfield on Saturday, with Mohamed Salah and Philippe Coutinho injured and forward Roberto Firmino left on the bench.
"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," said Klopp.
UNDER THREAT
Lovren and Klavan, whose places will be threatened by the arrival of Virgil van Dijk for a reported 75 million pounds ($100.49 million) from Southampton, combined to grab the winner, leaving seventh-placed Burnley without a victory in five games.
But the home side's manager Sean Dyche said it was important to maintain sight of the broader picture.
"On and off the pitch we have moved forward in a calendar year. We are becoming assured of ourselves in this division," he said.
"We are playing a more mixed game now. Last year we got it (the ball) forward more often but we are calmer now. We are resolute defensively, we have added to what we had. The quality has risen.
"To be on 34 points and be unfortunate to not have 35 shows what a season we have had," added Dyche.
(Reporting by Simon Evans; editing by Ken Ferris)
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