It was no more than the Blue Toon deserved after a controlled performance which saw Jamie Buchan grab a double.
The Rovers pitch passed an earlier inspection, but both sides struggled to find their feet on the heavy ground for most of the half.
However, it was Albion who had the first chance to take the lead after 12 minutes when a mistake by Robbie Raeside in the Peterhead defence let Marc McKenzie through on goal, but his shot flew just past the post.
This sparked the Blue Toon into life and they took the lead in the 23rd minute when Buchan played a neat one-two with Graeme Stewart on the edge of the box and fired an unstoppable shot into the top corner.
Midway through the half the game was held up for a few minutes with a bad head injury to Rovers' defender Jed Stirling, and the rest of the half was played out rather tamely.
Then seconds into the second half Peterhead extended their lead as Craig Campbell crossed the ball for Robbie Linn to glance a header in off the underside of the bar.
Peterhead were very much in control by this point, and they made it three after 58 minutes when yet again Campbell got the ball down the right, fired the ball across the face of the goal and Martin Bavidge was on hand to prod the ball over the line.
Albion had a few chances in the second half to grab what would have been no more than a consolation as both Paul McManus and McKenzie went close.
However, it was Buchan who rounded off the game with a bizarre goal as he charged down a Danny Black clearance and saw the ball fly into the roof of the net.