Stoke scored again early in the second half through a close-range finish from Dutch midfielder Ibrahim Afellay on 52 minutes.
Before Bournemouth could regroup, they were 3-0 down when Stoke substitute Joselu, having only just come on, headed in on 55 minutes.
The hosts this time quickly hit back through half-time substitute Matt Ritchie in the 57th minute, but Stoke went on to close out a comfortable victory.
Struggling Norwich blew a 2-0 lead as West Ham fought back to draw 2-2 at Carrow Road.
The Canaries, who had lost five successive games to plummet into the bottom three, went ahead through a fine curling effort from Robbie Brady on 54 minutes before Wes Hoolahan doubled their advantage from close range on 65 minutes.
However, West Ham midfielder Dimitri Payet marked his new contract extension with a neat finish from a tight angle on 74 minutes before Mark Noble silenced the home supporters when crashing in a equaliser just two minutes later.
Southampton won 1-0 at Swansea with a header from Shane Long on 69 minutes to extend their unbeaten league run to six matches, and without conceding a goal.
Troy Deeney scored twice as Watford secured a 2-1 victory at Crystal Palace, who remain without a Premier League win since December 19.
The visitors went ahead on 16 minutes when Deeney converted a penalty.
Palace levelled in first-half stoppage time through a looping header from Emmanuel Adebayor, the much-travelled Togo frontman’s first Premier League goal since October 2014.