After striking the post with an early free-kick Coutinho embarked on a 40-yard run in the 20th minute, leaving several players in his wake before applying a low finish.
Six minutes later Dejan Lovren met Roberto Firmino’s acrobatic cutback with a diving header.
Jermain Defoe hit the post for the hosts but Salah made it 3-0 on 44 minutes after weaving his way in from the right for his 20th goal in 26 matches.
Alex Oxlade-Chamberlain hit an upright for Liverpool before Firmino headed in the Reds’ fourth from Coutinho’s 66th-minute cross.
The win lifts Liverpool a point above Arsenal in fourth place. AFC Bournemouth slip to 16th.
AFC Bournemouth: 23 Dec v Man City (A)
Liverpool: 22 Dec v Arsenal (A)