Westley, who masterminded successive Boro promotions, took charge at Deepdale in January and his move was vindicated with an inspired showing at the Lamex Stadium.
Stuart Beavon put Preston ahead early on, and although Greg Tansey quickly levelled matters, strikes from Jack King, Jeffrey Monakana and Nicky Wroe secured the points for the Lilywhites.
It took less than six minutes for Westley's side to silence the home fans as Beavon made the most of a mix up to fire emphatically past Steve Arnold.
Boro responded and squared the contest after 15 minutes following a short-corner routine which allowed Tansey to curl a stunner into the top corner of the net. But Preston restored their lead with 32 minutes on the clock when Monakana ran down the left before picking out King, whose strike ricocheted off skipper Mark Roberts and past Arnold.
The visitors made the points safe within the hour as Monakana's angled drive bobbled beyond Arnold before Wroe completed the scoring in the 72nd minute with a delicate curling effort from the edge of the box.