The Cherries, who beat Chelsea last week at Stamford Bridge, opened scoring as early as the second minute, after Junior Stanislas’ corner-kick floated into the top corner.
Marouane Fellaini equalized before the break, but Joshua King grabbed the winner in the second half.
Earlier in the day, Manchester City claimed a narrow 2-1 win over Swansea City at the Etihad.
Odion Ighalo scored the winner at the Stadium of Light, as Watford beat Sunderland 1-0, Norwich City held Everton to a 1-1 draw, while there were no goals between West Ham and Stoke City at Upton Park.
