The midfielder's all-action display, rounded off by his first league strike since 2011, moved the Latics away from the relegation zone and left Boro rooted to the foot of Sky Bet League One.
The visitors almost grabbed an early lead when Jon Ashton's header came back off a post.
But Wesolowski settled a scrappy game after 13 minutes, meeting Mike Petrasso's cross with a neat eight-yard header.
Stevenage had Ben Chorley sent off for violent conduct in the 86th minute following an off-the-ball clash with Clarke-Harris.
Oldham manager Lee Johnson said: "This is a big win for us and James Wesolowski was fantastic.
"He was excellent last week at Liverpool in the FA Cup and you can see what a difference he makes.
"It isn't just that he drives us on, he makes so many tackles and is a real winner. He has got better technically this season and he deserved his man-of-the-match award."
Johnson went on: "Overall it wasn't a great performance. We didn't play free-flowing football, but the points are really important.
"We showed quality at times and had three or four chances, but it was edgy because this was a game we couldn't lose.
"Stevenage are big lads who spend a lot of time on the weights, but they have mobility and are difficult to play against.
"I'm pleased with the clean sheet because we've had nothing like enough this season and, with our games in hand, we've given ourselves some breathing space. It looks a bit more comfortable now."