With top scorer Mohamed Salah and playmaker Philppe Coutinho injured and Roberto Firmino on the bench, the seven changes Klopp made left his side short of ideas, despite Adam Lallana’s first league start of the campaign.
Salah, Coutinho, Firmino and Mane have scored 58 of Liverpool’s 77 goals this season – 32 of 48 in the Premier League – so their absence from the starting line-up at the ground which has witnessed only six away goals all season was always likely to be a risky strategy.
Alex Oxlade-Chamberlain’s long-range shot which forced Nick Pope to parry away and Dominic Solanke’s heavy touch in the penalty area were the closest Liverpool came to decent chances with most of those falling to the opposition.
Lallana denied Gudmundsson a clear run on goal as Burnley countered from a Liverpool corner, with Scott Arfield seeing one weak header saved and a low, left-footed shot fly inches wide of the far post.
Early in the second half Mane wanted a penalty for Phil Bardsley’s challenge, but referee Roger East waved play on.
On the hour Can picked out Trent Alexander-Arnold wide on the right and Mane controlled his deflected cross with one touch before smashing a left-footed effort into the roof of the net with his second.
Pope then tipped away an Alexander-Arnold drive as the visitors finally found a second gear, but Burnley’s threat was not diminished with Ben Mee heading over a corner and Arfield flashing another shot narrowly wide.
The improving Oxlade-Chamberlain had a near-post effort batted behind by Pope, with Simon Mignolet saving substitute Sam Vokes’ low header as the game remained more open than Klopp would have liked.
And so it proved as Vokes flicked on a cross for Gudmundsson to equalise, only for Klavan to snatch it back as Klopp’s unbeaten record was extended to 16 matches – his best run since coming to England.