James 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.
"Before today we needed 10 more wins to make sure we stayed in this league, and that remains the case," said Boro boss Westley.
"It's a tough ask and we haven't had the best first half to the season, but we can't dwell on that.
"Fourteen of our 24 league games have been away from home and everyone knows it's tough to travel. Home advantage is exactly that, so it's important we capitalise. We have to play like the team we can be, instead of playing like our league position.
"This was an even game and both goalkeepers were dramatically under-worked.
"We shaded possession, but there weren't many chances and neither side did enough to win.
"Oldham were determined to hang on to their goal and they defended well, which we knew they could do.
"We struggled to carve out opportunities, but there was nothing in it."
Westley added: "I didn't see the incident for the red card and the footage isn't clear, so I can't really comment. But we have struggled all season for available defenders and it looks like Ben Chorley will miss three games."
- Sports & Recreation
- Ben Chorley
- Dean Parrett