Tottenham, Chelsea's only realistic title rival, applied pressure by defeating Watford 4-0 hours earlier at White Hart Lane.
But Chelsea got going early through a fortuitous own goal off a Diego Costa shot, and Eden Hazard's calm finish made it 2-0.
Josh King pulled one back for Bournemouth before the break, only for Marcos Alonso's stunning free-kick to seal victory for Chelsea.
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AFC Bournemouth: Boruc, Adam Smith, Francis, Steve Cook, Daniels, Fraser, Arter, Wilshere, Pugh, King, Afobe
Subs: Gradel, Brad Smith, Lewis Cook, Allsop, Mousset, Ibe, Cargill
Chelsea: Courtois, Azpilicueta, Cahill, Luiz, Alonso, Moses, Matic, Kante, Pedro, Costa, Hazard.
Subs: Begovic, Fabregas, Zouma, Willian, Batshuayi, Terry, Chalobah
Referee: Andre Marriner (West Midlands)
WHAT THEY SAID
Antonio Conte: "This race is open. I think now I can see two teams, Chelsea and Tottenham, in this race.
"But anything can happen because there are 24 points (to play for) and other rivals can go in this race."
Eddie Howe: "We are feeling good and confident at the moment.
"Our home form has been very good, we have won our last two here and performed well away in the last two as well, against Southampton and Liverpool, which were really big points for us.
"If we can continue that momentum we have a chance."