Ishmael Miller sealed victory for promotion-chasing Boro in the npower Championship encounter at London Road by slamming in a 76th-minute winner, just three minutes after his introduction.
Mowbray said: "We came here to win, we would have been disappointed to draw and yet we could even have lost.
"It was all about the three points and we got them in the end. We set up to play on the counter-attack and after 25 minutes it looked like the right tactic but we didn't go on to dominate the game in the way we should.
"A fantastic goal brought Peterborough back into it and a sloppy one got them level, but I'm thankful our changes worked well. Ishmael was a real thorn in Peterborough's side. He was unlucky not to play from the start but showed his hunger to help the team by coming on and doing the business.
"The fact that Peterborough are bottom shows just how strong the Championship is as a league."
Posh boss Darren Ferguson felt his side could have gone on to claim victory, but they ended up being beaten for the ninth time in 11 home games thanks to Miller's goal and Olejnik's error.
Ferguson said: "Tony is an honest man and he will know his team have been fortunate to leave with a win. When we equalised I felt sure there would only be one winner, but unfortunately it wasn't to be.
"There's no doubt in my mind we deserved to win it but we have conceded three very poor goals - one from our own corner, one when we have failed to pick up a man at the back post and the third from a goalkeeping mistake.
"But there are a lot of positives to take from our performance and we'll certainly win games if playing to that standard on a regular basis. We have 25 matches to turn our season around."