Pressured by Tottenham’s earlier thrashing of Watford, the visitors looked to be cruising after two goals in three first-half minutes.
But Bournemouth, who had already hit the post, got back into the contest through Joshua King’s long-range deflected effort.
The Cherries continued to press after half-time only for Chelsea to gradually regain control and secure the points through Marcos Alonso’s impeccable free-kick.
The Blues next travel to Manchester United, their sternest of six league fixtures between now and the end of the season.
If they return from Old Trafford with their seven-point advantage still intact, they will be a huge step closer to a sixth title and fifth in the Premier League era.