The hosts looked to have earned a point when Johann Gudmundsson equalised with just three minutes of normal time remaining.
But Klavan headed home from almost on the goalline as the Reds snatched all three points.
Sadio Mane had given Liverpool the lead earlier in the second half with a superb shot from the edge of the area.
Burnley goalkeeper Nick Pope made several good saves to keep his side in the match, while opposite number Simon Mignolet denied Sam Vokes just before Gudmundsson’s equaliser.
Victory moved the fourth-placed Reds within a point of third-placed Chelsea on 44 points, while Burnley stay seventh.
Liverpool have not lost in their past 13 Premier League matches, but the Clarets are on a run of five without a win.
Klavan and defensive partner Dejan Lovren could find their places under threat following the arrival of Virgil van Dijk from Southampton for £75m.
So their combination for Klavan’s 94th-minute winner, with Lovren providing the assist with a flicked header, was timed to perfection.