The visitors dominated the early stages and were rewarded by taking the lead after 22 minutes, James Wesolowski running in to score with a shot from 10 yards after Clarke-Harris had headed the ball across the face of goal.
Bradford equalised after 35 minutes when a 30-yard pass from Matthew Dolan found on-loan winger Adam Reach in the penalty area and he scored with a low shot into the far corner.
But Oldham equalised on the stroke of half-time when Gary Harkins drifted in from the left before sliding the ball into the path of the unmarked Clarke-Harris to score from the edge of the six-yard area.
Oldham increased their lead after 58 minutes when Clarke-Harris ran on to a long through pass from left-back Jonathan Grounds and held off the challenge of defender Andrew Davies before shooting low into the far corner for his second goal of the match.
Skipper Gary Jones pulled a goal back for Bradford in stoppage time with a low shot from the edge of the box after substitute Mark Yeates had hit the post, but it was too little, too late for the home side.