Rovers started with just one man up front and their defensive plan was almost undone in four minutes when Ryan Blackadder's 30-yard free-kick landed on top of the net.
But they stunned the league leaders in the seventh minute when Bryan Felvus robbed Jonathan Smart, and squared to give Tommy Lennox an easy tap-in.
East Fife levelled on ten minutes when Jon Smart's header was nodded off the line but Kevin Gordon drove the rebound through a ruck of players and into the net.
Rovers went ahead again on 18 minutes when a cross was poorly cleared and Phillip Creaney volleyed the ball into the bottom corner.
Jamie Ewings kept Albion ahead with excellent saves from both Jon Smart and Craig Smart inside 60 seconds.
Rovers threatened on the break and John Dodds took two attempts to hold a Creaney drive on 43 minutes.
Craig Smart was then denied by Ewing's foot and Blackadder crashed a shot off the bar as Albion held on.
The Fifers managed an equaliser when Craig O'Reilly headed in Jason Dair's cross on 56 minutes.