Albion where first to threaten in four minutes when a Bryan Felvus free-kick flew just wide of the post.
Montrose settled down and in 13 minutes Paul Napier shot wide from long range then a 16-yard header from the same player was held by keeper Jamie Ewings.
On 21 minutes a free-kick from Jered Stirling was blocked by the Albion wall.
Montrose were out of luck in 26 minutes when Stuart Cumming linked with Andy Rogers and smashed a shot off the post.
Three minutes later a chip from Rodgers was tipped away by goalkeeper Jamie Ewings.
Two minutes from half-time a drive from Scott Friel was well saved by home keeper Andy Reid.
Right on half-time a chip from Rodgers dipped just wide of the Albion post.
Montrose opened the scoring in 51 minutes when Gordon Lennon fouled Scott Michie in the box and Rodgers scored from the spot.
Montrose stretched their lead in 58 minutes when Rodgers was fouled by goalkeeper Ewings and converted his second penalty of the game.
Montrose looked for more goals and in 74 minutes a Michie cutback was met by Greig Henslee and keeper Ewings managed to block.
Albion gave themselves a lifeline seven minutes from time when Andy Sim drove the ball into the net through a crowd of players.
Albion could have been level just two minutes later when Felvus had an effort cleared off the line by Chris McLeod.