A classy 69th-minute shot by substitute Leandro Trossard sealed the three points and lifted the level of quality against a limited Everton team who were playing spoilers at home.
A third straight 1-0 loss at Goodison Park this season suggested Everton's prospective new American owners would be taking on a rather distressed asset on the field.
And it sets up next weekend's north London derby between Arsenal and Ange Postecoglou's Spurs perfectly.
It was Everton's first game since the club announced majority shareholder Farhad Moshiri had agreed a sale to Miami-based 777 Partners.
777 has stakes in multiple clubs worldwide including Hertha Berlin who, like Everton, were in decline on the field when a majority stake was taken in March and have since been relegated.
Chelsea are still misfiring with yet more newly acquired players in a $US1 billion ($A1.6 billion) spending spree and after hiring another highly-rated coach since the American-led takeover last year.
Coach Mauricio Pochettino was missing a full team's worth of injured players and filled his subs' bench with teenagers yet to make their first-team debuts.
Still, his team had enough scoring chances at Bournemouth and twice had shots strike the frame of the goal.
"That is a thing that makes me very, very angry and disappointed because the effort of the team is fantastic," Pochettino said.
"We need, sometimes, this luck or be more clinical in front of goal. Because all the metrics are good."
The luck was missing when Raheem Sterling's 51st-minute free kick bounced down from the bar on to the goal line. Though defender Levi Colwill followed up to put the ball in the net, he was ruled offside.
Chelsea now have just one win in five league games and are 14th in the standings – one place below Manchester United and one above Bournemouth who have yet to win.