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Jurgen Klopp’s side had taken the lead just after the hour after a fine strike from Senegal forward Sadio Mane -– his eighth goal of the season — but Johann Berg Gudmundsson levelled three minutes from time with a diving header.
However, Klavan — the first Estonian to score a goal in Premier League history — ensured Liverpool clinched all three points when he nodded in from close range in the fourth minute of added time.
Liverpool had scored 30 goals in eight away games in all competitions before this clash, but found Burnley more stubborn opposition.
For the majority of the game the Reds were required to show a grittier side to their play compared to the free-flowing football seen this season.
Klopp made seven changes from the side which came from behind to clinch a 2-1 win over Leicester City 48 hours previously.
The visitors were without in-form duo Philippe Coutinho and Mohamed Salah, who had thigh and groin injuries respectively.
Liverpool were fortunate not to be punished early on when Trent Alexander-Arnold was caught by Scott Arfield who slipped in Ashley Barnes, but the Clarets striker lifted his shot over the crossbar.
The Reds found their rhythm and a intricate passing move culminated in a shooting opportunity for Alex Oxlade-Chamberlain, who saw his low angled drive beaten away by Burnley goalkeeper Nick Pope.
Liverpool dominated possession as the first half wore on but without Coutinho’s quick-thinking creativity in midfield they struggled to break Burnley down.
Burnley threatened sporadically as they looked to soak up the pressure, although Simon Mignolet was called into action to save Arfield’s glancing header.
Without Coutinho and Salah this was a game when Liverpool needed Mane to step up and recapture his scintillating form of last season, and despite largely struggling he provided a moment of quality.
When Alex Oxlade-Chamberlain intelligently picked out the run of Alexander-Arnold on the right flank, the full-back’s deflected cross found Mane in the centre of the box.
The Senegal forward controlled the ball turned on sixpence and lashed an unstoppable shot past Burnley keeper Nick Pope and high into the roof of the net.
Liverpool looked the more likely to score a second and a brilliant save from Pope kept out a thumping shot from Alexander-Arnold which looked destined for the top-right corner.
Burnley equalised in the 87th minute when Charlie Taylor’s cross from the left was flicked on by substitute Sam Vokes and the stooping Gudmundsson headed past Mignolet.
However, Liverpool scored with the seconds ticking away when Oxlade-Chamberlain’s free-kick was headed back across goal by Dejan Lovren to leave centre-back partner Klavan with the simplest of headers from a few yards out.
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