The Robins have a terrible away record in the league this season, with just one point from eight games, but they produced a dogged defensive effort against an Oldham side who have been strong at Boundary Park.
Goalkeeper Scott Brown led the way as Town fought bravely for their clean sheet when they came under massive pressure in the second half.
Oldham missed their first-choice strikeforce of Darren Byfield and Lee Hughes, who were both unavailable, and a series of chances went begging and it is Martin Allen's strugglers who will go to Morecambe in the second round.
They could have gone in front in the first five minutes when Lee Ridley and Ashley Vincent both brought saves from home keeper Mark Crossley.
The breakthrough arrived midway through the first half when Montrose was given too much space eight yards out and drilled his shot low past Crossley.
Oldham toiled up front and created very little until defender Stefan Stam headed goalwards and Brown tipped his effort away.
It was all Oldham after half-time, but substitute Matthew Wolfenden was denied by Brown with the first of many openings.
Brown also denied Lewis Alessandra and Kieran Lee, whose next effort flashed inches wide of the far post.
Cheltenham pulled every man behind the ball in the closing stages, which included seven minutes of injury time, and their backs-to-the-wall attitude pulled them through.
They had a major scare three minutes from time when Alessandra was thwarted by Brown and the same striker then missed his kick from six yards out.
Wolfenden missed the final chance before the Latics were booed off by their own supporters.