Westley was sacked as manager after 13 months in charge at Deepdale in midweek with the side having won just once in League One since November.
But Stuart Beavon lifted the gloom with a fine strike after 19 minutes and Bailey Wright doubled the lead before the break to dent the Cherries' title push.
Bournemouth's Brett Pitman and Preston's John Welsh both enjoyed sights of goal before Beavon gave the Lilywhites the lead, rifling a low 20-yard strike past Bournemouth goalkeeper Shwan Jalal.
Preston then doubled their advantage after 31 minutes when Wright bundled in Nicky Wroe's free-kick from two yards.
Lee Holmes broke free of the Cherries' midfield just before the hour, but his 20-yard drive flew wide of the right post.
Steve Cook headed Matt Ritchie's cross harmlessly over after 66 minutes and Pitman then had the chance to grab one back with 10 minutes remaining, but made minimal contact with Ritchie's centre.