Klavan's close-range header in the fourth minute of added time made light of the absence of three of the 'Fab Four'.
With £75 million world-record defender Virgil van Dijk waiting in the wings for his debut, the focus has been turned on the Reds' central defenders, but it was at the other end of the pitch where Estonian Klavan proved his worth.
Klopp conceded afterwards that the result was a lottery: "It was the most difficult circumstances for both teams, I would say, playing 48 hours ago. Both teams had an opponent which was motivated on the highest level, ready for these difficult circumstances with rain, wind and cold for the players.
"You need first of all concentration, readiness to work really hard, you need good organisation, discipline and stuff like that – and, of course, a little bit of luck. We deserved the three points but a draw would have been absolutely OK as well and if we don't score then even Burnley would have deserved it, it was just an open game. We had more of the ball, we played more, but the main job was to deny all the strengths of Burnley. When they win the ball, they are so tuned in these balls in behind – kind of channel balls – and Barnes running, Hendrick running, it's really difficult to defend.
"We wanted to have a high last line to help a little bit, but that means we need to be even more concentrated. As long as the game is open, one goal can be the winner. It looked for a while like that until Burnley scored and then we scored pretty much the twin of that goal, very similar.
"It is just difficult to defend when you lose the first header, which we did and which Burnley did, it's an open ball in the six-yard box and is really difficult. You need to be there – and both teams were there. We were the lucky side, but – as I said – I am really happy about attitude, character and all that stuff."