Brett Emerton hit the winner for Rovers in a keenly fought contest.
Allardyce said: "You know there is no time for the opposition to come back then. It was our most difficult period of the game because they were right up for it. We were coping with everything they threw at us but then they got the goal and that lifted them.
"They started throwing more and more people forward to get the winner and left more and more spaces at the back. In this league counter-attacking football is a major part of your armoury and we caught them on the break.
"Brett Emerton came up with an outstanding finish and that was it, we had won 2-1.
"We were fortunate with the first goal but after that we were comfortable. We have got a very valuable three points and are very pleased to get our first away win."
Victory was especially sweet for Allardyce, coming at the club which sacked him 14 years ago.
He added: "You always want to do well against one of your old clubs if you've managed them, but it was a long time ago. It is great to see them up here. It is a bit of a fairytale for them."