The Reds had gone in front thanks to a superb solo goal from Mane but saw Johann Berg Gudmundsson equalise in the 87th minute before Klavan nodded in at the death.
Reds boss Jurgen Klopp made several changes:
By contrast, Burnley's lineup had a more familiar feel:
Klopp's rotated starting XI struggled to get up to speed during a first half lacking in pace, quality and intensity. While Liverpool laboured to get the creative juices flowing, the hosts were content to sit deep and play on the counter.
It was no surprise when a drab 45 minutes shorn of chances ended goalless.
The patter of play barely changed after the restart, as Liverpool missed the guile of playmaker Philippe Coutinho, who was not included due to a thigh problem, per Rory O'Callaghan of Sky Sports.
As much as Coutinho's influence on Liverpool's passing was missed, the visitors also looked lost without the outlet provided by Mohamed Salah's pace. The Egyptian winger was also out with injury, according to David Maddock of the Daily Mirror.
Fortunately, Liverpool can still count on Mane for pace and perceptive movement. It was the Senegalese winger who got on the scoresheet just after the hour mark thanks to a terrific shot into the top corner.
The goal encouraged Burnley to show more intent and midfielder Steven Defour came close to an instant equaliser when his low shot forced Simon Mignolet into a decent save.
Knowing his side needed more up top, Clarets manager Sean Dyche replaced midfielder Jeff Hendrick with striker Sam Vokes. Meanwhile, Liverpool hooked goalscorer Mane for Roberto Firmino.
Vokes' introduction gave Burnley more impetus to go direct, and his knockdown led to strike partner Ashley Barnes volleying inches wide. Next, it was Vokes who drew Mignolet into a solid stop.
Burnley's pressure was beginning to tell, and the home side was level two minutes from time. The goal came courtesy of a diving header from Gudmundsson after another useful touch from Vokes.
It should have been enough to seal a point, but Klavan had the last word when he reacted quickest to a flick from fellow centre-back Dejan Lovren.
Winning without Coutinho and Salah makes a statement about the improving strength in depth of Klopp's squad. Liverpool now look like firm favourites to finish in the top four for a second season running.