The Reds trailed after just 12 minutes at Anfield as Jonathan Walters powered home a header from Erik Pieters' left-wing cross.
But that blow only seem to bring Jürgen Klopp's team to life, and they were ahead by half-time thanks to Adam Lallana and Roberto Firmino.
A Giannelli Imbula own goal and a further strike from Daniel Sturridge added deserved gloss to the scoreline in the second period - and provided reward for the home team's unshakeable belief, according to Klavan.
"It was nice to see the whole team adapted to the game," the defender told Liverpoolfc.