James Wesolowski's goal - his first in the league since 2011 - settled a scrappy game and left Boro rooted to the bottom of League One.
The visitors went close to a ninth-minute breakthrough as Jon Ashton headed against a post from a corner.
But it was Oldham who struck after 13 minutes when Mike Petrasso crossed from the right and the unmarked Wesolowski headed home from eight yards out.
Latics goalkeeper Mark Oxley did well to deny Filipe Morais, while Stevenage striker Francois Zoko missed a good chance on the volley.
Jonson Clarke-Harris fired off target either side of half-time for the hosts before Oxley made a smart save to thwart Dean Parrett.
Defender Ben Chorley received a straight red card for violent conduct in the 86th minute following an off-the-ball clash with Clarke-Harris.