Henderson curled a teasing cross into the penalty area and Mane found a pocket of space inside the Bournemouth area to glance a header at goal which goalkeeper Asmir Begovic parried.
James Milner (NA): Came on for Sadio Mane, who looked a bit jaded.
Third-from-bottom Southampton are five points from the safety zone after relegation rivals Crystal Palace beat Brighton & Hove Albion by the same score at Selhurst Park and Huddersfield Town beat Watford 1-0.
Salah should have doubled Liverpool's lead on 33 minutes but was denied by Cook with a superb block. The ball carried about 30 yards before nearing the Liverpool star.
Salah, 25, endured a miserable stint in England back in 2014 when at Chelsea, but was soon sent to Italy.
And Mane who opened the scoring before Mohamed Salah and Roberto Firmino added the remaining has disclosed how the German tactician got them psyched up for another impressive display. Liverpool midfielder Alex Oxlade-Chamberlain dribbled down the middle of the pitch and laid off a pass to Firmino just as he reached the box.
"We are a team and we have got to play together and if he does it he does it, we will all be happy for him, but I don't think we are going to focus too much on it".
The important thing is that we know we depend on ourselves. "It's like that but in the end we play as a team".
Liverpool face Roma in the semi-finals of the Champions League and ex-Reds stopper Bruce Grobbelaar is tipping them to go the whole way.
It continued to be one-way traffic in the opening 45 minutes and Begovic denied both Alexander-Arnold and Salah before Bournemouth finally threatened. It's also a different season for him. It's completely normal that a boy in that situation wants to be top of the scoring list, the Golden Boot.
- MI vs DD IPL 2018 Match 9 Live Streaming
- Cardi B's 'Invasion of Privacy' Debuts at Number One on Charts
- Starbucks manager leaves company after calling police on two black men
- Expect wet, windy weather for the north valley's Monday
- Man arrested, charged with domestic battery with 2-year-old present
- West Brom striker Rodriguez cleared over FA charge
- England win netball gold at Commonwealth Games
- Russian Federation calls an emergency session to discuss strikes on Syria
- Sean Hannity Was Trump Lawyer Michael Cohen's Mystery Client
- IPL 2018: Gambhir returns to Kolkata as skipper of Delhi