The winning goal came courtesy of debutant Joe Bunney's first touch in professional football, the young striker heading home in the 78th minute shortly after coming on from the bench.
The Pilgrims' quest suffered a serious setback 14 minutes in when Maxime Blanchard was dismissed for a foul on George Donnelly.
The Rochdale striker had lifted the ball over Blanchard's head and was advancing towards goal when the defender hauled him down, referee Geoff Eltrincham deeming it a red-card offence.
Rochdale took control after that and Jake Cole pulled off good saves to deny Donnelly and Reece Gray, while Michael Rose's 35-yard drive smashed against the underside of the crossbar.
The visitors saw their appeals for a penalty waved away when Wayne Thomas appeared to foul Joe Bryan, but the home side continued to ask most of the questions.
And they clinched the win when, minutes into his debut, Bunney headed home fellow substitute Scott Tansey's cross from six yards.