Adam Smith’s own goal, Eden Hazard’s 14th of the season and a Marcos Alonso free-kick restored their seven-point lead over Tottenham Hotspur, who had started the day with a 4-0 win over Watford.
Chelsea led on 17 minutes at Vitality Stadium when Diego Costa's off-target shot hit Smith to go in.
Three minutes later it was 2-0 as Hazard raced on to N’Golo Kante's through-ball and rounded Artur Boruc.
Joshua King's deflected shot gave the Cherries hope before half-time.
But Chelsea sealed the points thanks to Alonso's free-kick midway through the second half.
The defeat drops AFC Bournemouth to 15th, seven points above the relegation zone.
Man of the match Marcos Alonso (Chelsea)