It looked like being a frustrating afternoon for the Reds at Turf Moor as they entered stoppage-time with the game locked at 1-1.
Klavan, though, was to have the final say with a dramatic header which snatched all three points for Jurgen Klopp’s side.
In finding the target, the Estonian defender has become the first player from his country to net in the English top-flight.
1 - Ragnar Klavan's goal for Liverpool is the first scored by an Estonian player in Premier League history. New.— OptaJoe (@OptaJoe) January 1, 2018
97 - Estonia become the 97th different nation to have a player score a Premier League goal (excl. OGs). Global. https://t.co/Gr6wfmJE5d— OptaJoe (@OptaJoe) January 1, 2018
Liverpool had taken the lead at surprise package Burnley just past the hour mark through Sadio Mane.
They were then pegged back in the 87th minute through Johann Berg Gudmundsson.
With time ticking away, the Reds found a winner as Dejan Lovren flicked on a free-kick and his centre-half partner stooped at the back post to nod home.
A battling victory ensures that Liverpool open 2018 inside the Premier League’s top four, with a New Year’s Day success edging them six points clear of fifth-placed Arsenal – although the Gunners do have a game in hand.