The win means Liverpool remain unbeaten after 16 league games, securing 15 wins and one draw.
They have now temporarily gone 11 points clear of second place Leicester City, on 46 points.
Liverpool were forced into an early substitution as Dejan Lovren went off injured and was replaced by Trent Alexander-Arnold on 39 minutes.
After holding out for most part of the first half Bournemouth’s resolve was finally broken on 35 minutes as Alex Oxlade-Chamberlain finished off Jordan Henderson’s superb ball over the top.
Liverpool doubled their lead on 44 minutes through Naby Keita who finished coolly off a lovely backheel assist from Mohamed Salah.
In the 54th minute Liverpool went 3-0 ahead as Keita this time turned provider by feeding an inch-perfect through ball to Salah who made no mistake.
In London, Tottenham Hotspur put their disappointing defeat to Manchester United in midweek as they hammered Burnley 5-0.
Spurs got off to a very good start as they took the lead on five minutes off Harry Kane whose long range strike swerved in past Nick Pope.
Just four minutes after Kane’s opener, Lucas Moura made it 2-0 poking in at the far post after a goalmouth scramble.
Spurs continued to create chances and deservedly went 3-0 up on 32 minutes through Heung-min Son who picked up the ball deep in his own half and ran all the way through to score.
Kane grabbed his second goal nine minutes into the second half as he curled his effort past Pope following Dele Alli’s assist.
Spurs continued to pour forward and were on target again on 74 minutes through Moussa Sissoko who playd a classy one-two with Kane and fired into the back of the net.
Spurs are now fifth on 23 points and sit six points away from fourth position on the log.
And at Vicarage Road, Nigerian forward Isaac Success was benched by new manager Nigel Pearson in Watford’s 0-0 draw against Crystal Palace.
Success have only played three Premier League games this season without scoring.
Watford have now gone four consecutive league games without a win and remain bottom on eight points.
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