Rovers went into the game hoping to erase the memories of a 5-1 home drubbing at the hands of Oxford United last time out, that defeat having brought a five-game unbeaten run to an emphatic end.
Boss Jim Harvey made two changes, with Mark Beesley stepping down to the bench and Allan Russell suspended - Les Afful and Adriano Rigoglioso earned starting spots.
The Brewers arrived at the New Lawn hoping to maintain the form that had seen them go nine games without defeat since a 1-0 reverse to Southport in late August.
Nigel Clough made two changes from the side that drew 0-0 at Woking last time out, Daniel Holmes and Shaun Harrad taking the places of Gary Rowett and Steve Scoffham.
Rovers were first to threaten, with Simon Clist testing veteran Albion keeper Kevin Poole with a stinging shot from just outside the peanlty area in the third minute.
However, both sides struggled to carve out openings as the game progressed up to the 20-minute mark.
The visitors then got into their stride, with Steve Williams making an excellent save to tip a Lee Fowler drive away for a corner 26 minutes in, before Daryl Clare again tested the home keeper.
However, just past the half-hour mark, Rovers were in front as Rigoglioso netted his third goal in four games since joining on loan from Morecambe.
The midfielder tested Poole seconds before he found a way through to break the deadlock, following good work by Alex Lawless and Afful.
Williams then ensured his side went in a goal up at the break as he denied Chris Hall with another quality save eight minutes before the interval.
The second half started as quietly as the first had, before visitors' skipper Darren Stride was booked for a foul on goalscorer Rigoglioso.
Brewers boss Clough then decided to replace Hall with striker Jon Shaw just before the hour mark to try an pep his side up.
Michael Brough fired just over the crossbar from 20 yards as Rovers went looking for a second goal to ease their nerves, before Rigoglioso was replaced by Beesley.
Meechan forced an impressive save from Poole with 18 minutes left to play, before Harvey threw on Danny Williams in place of the tiring Afful.
Brough headed over from a Clist corner as time ticked away, shortly after Shaw had met an Andrew Corbett cross, only to send his effort straight at Williams.
Harrad and Aaron Webster threatened late on, before both sides hit the woodwork in time added on at the end of the game.
First sub Paul Stonehouse's shot rattled the Brewers crossbar, before Webster was denied a last-gasp leveller when his left-foot drive from the edge of the box struck the upright with Williams beaten.