Klavan popped up to net his second ever goal for the club and put the pressure on the other sides competing for a Premier League top-four finish.
Reflecting on a crazy end to the game, he told Liverpoolfc.com: "It's not about the goal but it feels great to have three points today.
"It was a crazy game in the end. That we took those three points with us is massive.
"I said it already when I came here that we have such a great team. Today we showed it again, that we have a great team and we keep on going.
"Of course, it's never easy here at Burnley. If you take also that the last game was only two days ago, some changes as well but everyone is ready and everyone stepped up and in the end three points."
The centre-back went on to pay tribute to the travelling fans at Turf Moor, adding: "It's massive support again.
"Thank you for the drive up here on January 1; it's a good start of the year for everybody."