Rovers took the lead when Tim Howard fumbled the ball and allowed Nikola Kalinic to step in and show his quick thinking and skill to put the ball into the back of the net.
Allardyce told the BBC: "It was a mistake but the ability he showed to capitalise on it was there for all to see.
"It's only the first game of the season but it's a very precious three points and the first three are as important as the last three."
He continued: "The defending of the team as a whole was exceptional, making sure we kept hold of what we got.
"We knew Everton would have to come back into the game at some stage, but we didn't get the second goal when we had the chances to do that. We missed golden opportunities to get the second goal so we expected a bit of pressure at the end, but we held on well.
"We were the better side because we created more clear-cut chances than Everton."