Shrews midfielder Paul Parry found the top corner after four minutes to score the games' only goal as Preston failed to mount a comeback.
The visitors enjoyed plenty of possession but did little with it against a well-drilled home defence, with Nicky Wroe's powerful 70th-minute free-kick as close as the Deepdale men came to a leveller.
Westley said: "Having gone behind, we dominated play in the first half and created some good moments but didn't take our chances.
"In the second half it was a bit backs-to-the-wall from their point of view. We kept the play going forward but didn't do enough to score the goal that would have got us back in it.
Westley added: "This is a tough place to come. They know how to play their ground, know how to play the officials and they're a tough nut to crack here."