But a nervy win following three defeats in four games stretched Liverpool's lead at the top to 25 points, with just three more wins needed to win their first title in 30 years.
Klopp told the BBC: "I know what the boys are able to do, but I am not the one that is playing, they are. We have said before, confidence is not something that you get, put in your pocket and keep for the rest of your life. It comes and goes.
"You need that feeling where things work out.
"You have to fight hard, that's always the main thing to do, but we have 82 points now, which is really nice. We have to carry on fighting. Give everyone a proper fight and that's what we did today."
It briefly looked like Liverpool's poor run was going to continue when Bournemouth's Jefferson Lerma rolled the ball across the goal for Callum Wilson's right footed shot to slide under Adrian after nine minutes.
VAR replays showed Wilson pushing Joe Gomez in the lead up to the goal, but despite protests from Klopp and his players it was allowed to stand.
It didn't take Liverpool long to bite back with Mane and Salah pairing up to bring the score to 1-1 on 25 minutes. Mane took the ball off Jack Simpson and despite a poor pass behind Salah, the Egypt international checked back inside and drove a left footed shot into the bottom right corner.
Mane teamed up with Virgil van Dijk for Liverpool's second on 33 minutes. A fast break from Van Dijk left Mane unmarked and in space. The Senegal international pulled Aaron Ramsdale out of his goal and slipped a low right footed shot into the bottom right corner.
Despite their lead, Liverpool looked unconvincing in the second half. Bournemouth nearly equalised on 61 minutes when Ryan Fraser rushed past Liverpool's defence and looped the ball over Adrian's head. A rushing James Milner just managed to hook the ball on the line and clear it.
Mane and Salah could have both had second goals with Mane beating Ramsdale and hitting the top right post and Salah whizzing the ball across the goal 15 minutes from time.