A contentious ninth-minute opener from Callum Wilson gave the Cherries an advantage that was wiped out by the leaders in the remainder of the first half.
Mohamed Salah marked his 100th league appearance for the Reds with the equaliser and Sadio Mane’s clinical finish later put them in front.
Jürgen Klopp’s side dominated the second half, too – but did require a pivotal James Milner clearance off the line – and did enough to post an all-time record 22nd consecutive home win in the English top flight.
Team news
Adrian continued between the posts due to Alisson Becker’s muscle injury, while Milner was deployed at left-back and Alex Oxlade-Chamberlain started in midfield.