Nicky Ajose's second-half goal gave Posh all three points, leaving Boro bottom of the League One table and nine points adrift of safety.
Home goalkeeper Chris Day had to be at his best in the first half to keep Peterborough at bay as he first kept out Danny Swanson's shot before later twice denying former Boro midfielder Michael Bostwick.
At the other end, Bobby Olejnik saved well from Darius Charles just before the Posh finally broke through when Ajose got in front of his marker to put away Mark Little's cross.
Westley said: "Their goal came against the run of play - we'd had a very strong period of pressure where Darius had a great chance and their goalkeeper has made a good save.
"Before you know it, one long ball and they've got four attackers against our eight defenders, the ball managed to drop to one of theirs, get crossed by one of theirs and it's in the back of our net.
"You couldn't say that they had a sustained period of pressure where they took the game to us and in the second half we were by far the better team.
"If there was a side that was going to score it looked like it was going to be us.
"But it can happen when you're down there and you don't get the rub of the green with Luke (Freeman) putting the ball across an open goal and then Darius' chance."
- Sports & Recreation
- Darius Charles
- Danny Swanson