And in the 90th minute, Firmino made the points safe as he fired into the bottom corner to make it 3-0.
The goal for Salah continued his incredible debut season at Liverpool in which he has now scored a stunning 30 Premier League goals and 40 in all competitions.
They had appealed the Premier League Match Centre's call to hand Christian Eriksen the goal, with Kane arguing the Dane's free-kick had brushed his shoulder, and the overturning of the decision prompted Salah to respond with surprise on social media.
The victory rounded off a fine week for Liverpool, who booked a Champions League semi-final against Roma by beating Manchester City 5-1 on aggregate with a second-leg win at the Etihad Stadium on Tuesday.
"Those three have been wonderful, but this is a team effort", stated Klopp.
Salah could have scored after three minutes, prodding a great opportunity wide from Trent Alexander-Arnold's whipped cross, but the first goal came in the seventh minute.
Jurgen Klopp believes Liverpool's clash with Bournemouth represented the ideal way to wind down from their midweek Champions League exploits.
It was Nathan Ake, playing on the left side of a back three, that had the challenge of keeping Salah quite, the Dutchman doing fairly well in comparison to some more senior defender's that have faced Liverpool's star man this season.
There was a sense this might not be Salah's day when he failed, correctly, to win a penalty moments before his goal.
Liverpool took the lead after seven minutes through Mane before Salah managed to lob Bournemouth's Asmir Begovic with a header in the 69th minute.
A third would come for Merseysiders, with Roberto Firmino getting in on the act by firing inside the post.
"Our lads tried to stay in the game as long as they could and the second goal is probably the one that kills it".
"I really liked Alex Oxlade-Chamberlain's performance".