The game was goal-less at half-time before the Cherries scored three times in the space of 17 minutes.
First Jordon Ibe played a cool one-two with Callum Wilson who slotted home, before goal-scorer turned provider to set up Junior Stanislas at the near post.
Ake then rubbed salt in the wounds of his old team by tucking in the third – leaving Chelsea fourth behind Liverpool on goal difference in the Premier League.
Elsewhere £57m man Aymeric Laporte celebrated his Manchester City debut with a clean sheet as Fernandinho, Kevin De Bruyne and Sergio Aguero goals led the league leaders to a 3-0 victory over West Brom.
City now hold a mammoth 15-point advantage over second-placed Manchester United after their rivals’ slip-up away to Tottenham.
There was joy for another new boy on Merseyside as Theo Walcott scored a double for Everton, beating Leicester 2-1 at Goodison Park.
Walcott had the simplest of tap-ins from Gylfi Sigurdsson’s cross, and then latched onto Michael Keane’s header to slot home before half-time. Jamie Vardy’s penalty proved a mere consolation for Leicester.
Sam Vokes rescued a point for Burnley, bringing to an end a run of three Premier League defeats, in a 1-1 draw with Newcastle after Magpies skipper Jamaal Lascelles had opened the scoring.
Jack Stephens was the hero for Southampton as he cancelled out Glenn Murray’s strike in their 1-1 draw with Brighton, but the Saints remain in the dropzone.
And Javi Gracia marked his first EPL game in charge of Watford in a stalemate at Stoke.
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