Rovers grabbed the lead with their first real attack of the game after just four minutes when Scott Chaplain pounced to drive home a low angled shot from 12 yards.
The home side should have doubled their advantage moments later when Lee Sichi went in on goal, but after rounding Derek Jackson he over-ran the ball and the chance went begging.
At the other end Shire's Marc McKenzie saw an effort pushed behind by Rovers' keeper Gary McGlynn.
From the resultant corner, Steven Dymock saw his header blocked on the line by Steven Noble before McKenzie's rebound effort was tipped over by McGlynn.
But Rovers doubled their lead after 24 minutes when Chaplain drove home from close range after George Wallace had nodded down a Noble free-kick.
The home side almost netted a third just after the break, but Dereck Ure got back to head Stephen Donachy's shot off the line.
Wallace had a close-range header well held by Jackson, and with 10 minutes left Albion's Gordon Lennon thumped an effort against the crossbar.