Improving Rovers, who had won four of their last five matches before Saturday, almost held on for their first away clean sheet of the season at Edgeley Park.
But Stockport's Tanto Olaofe headed home the winning goal in the 83rd minute after he was left unmarked in the six-yard box.
The defeat was a little harsh on Doncaster, who held their own for large parts.
Scroll down for our player ratings.
5. Joseph Olowu - 6
One or two rash moments and was one of the last men for the goal, but helped limit Stockport to relatively little. Photo: HOWARD ROE
6. Owen Bailey - 7
Made some important blocks and interceptions and even burst forward from defence at one point in the second half. Photo: Bruce Rollinson
7. Luke Molynuex - 7
Had a nervy moment or two in defence marking Louie Barry but Stockport's danger man was eventually subbed. Another positive display in attack. Photo: HOWARD ROE
8. Ben Close - 8
Excellent in the tackle and crucial to the cause in possession. His best all-round display this season. Photo: HOWARD ROE