After a scrappy first half, Senegal winger Mane showed Liverpool can thrive without his star team-mates as he opened the scoring in the 61st minute. Mane collected Trent Alexander-Arnold's pass and hammered home from the edge of the area for his eighth Liverpool goal this season.
Burnley equalised with just three minutes left as Sam Vokes flicked on and Johann Gudmundsson pounced to slot home. But Liverpool's Dejan Lovren headed on a free-kick four minutes into stoppage-time and Estonia defender Klavan poked home from close-range.
Liverpool's third successive win extended their unbeaten league run to 13 games as they cemented their place in the top four.