Liverpool dominated the first half but threatened little until scoring in the 35th minute.
Shortly after AFC Bournemouth's Nathan Ake had gone off injured, Jordan Henderson's ball over the top was converted first time by Alex Oxlade-Chamberlain.
Naby Keita doubled the lead just before half-time, finishing off Mohamed Salah's backheel pass.
Keita repaid Salah nine minutes after the break, slipping in the Egyptian to make it 3-0.
Liverpool go to 46 points, 11 clear of Leicester City in second.
A fifth straight defeat leaves AFC Bournemouth in 15th on 16 points, a point above the relegation zone.
AFC Bournemouth: 14 Dec v Chelsea (A)
Liverpool: 14 Dec v Watford (H)