Preston right-back Jack King headed his side in front in the 33rd minute before David Cotterill hammered home a swerving 20-yard shot that took a deflection off the head of Byrom to draw the teams level.
Preston restored their lead in the 61st minute with Scott Laird heading home at the near post following another corner from Byrom and Beavon's clincher came with 18 minutes left.
Westley said: "It's a step forward for us because they are a tough side to beat at home.
"We've had to work hard to develop our game away from home and this was a very professional performance.
"It was a well deserved and convincing win and the fans can see that the players are playing with a bit of pride. Now it's important to keep on winning."