Second-bottom Scunthorpe made the more confident start and midfielder Andy Barcham squandered a good chance after 18 minutes, firing over from 15 yards when free.
Andy Robinson came close in the 37th minute for the hosts, floating a free kick over Steve Mildenhall's bar after he was fouled following a great break from Ash Taylor.
Robinson was at the heart of all Tranmere's best moments and seven minutes after the break, he forced Mildenhall into a fine save, with sub Stockton unable to follow up.
Rovers took a deserved lead just before the hour when Taylor rose to meet Robinson's corner, with Mildenhall making a fine save before Stockton followed up and headed home from a yard.
Barcham could have drawn the Iron level with 13 minutes remaining but he fired high and wide from 20 yards with Owain Fon Williams out of his goal after failing to gather the ball.