Antonio, a recent arrival from Sheffield Wednesday, followed up a debut strike against Blackpool on Saturday to win the tie at Prenton Park, despite Michael Mancienne’s red card after the break.
Owain Fon Williams denied Antonio from the edge of the box early on but he was helpless when the winger headed home from Chris Burke’s 12th-minute cross.
Antonio then struck the crossbar after rising to meet Eric Lichaj’s ball in before Michael Ihiekwe sent a header inches wide at the other end.
Mancienne was sent off eight minutes after half-time when a studs-up challenge on Jason Koumas earned him a second yellow card.
Tranmere defender Danny Woodards then rattled the Forest crossbar before Eliot Richards slid a 78th-minute shot agonisingly wide for Rovers.
But that was as close as they came to an equaliser, with substitute Cole Stockton firing over the bar late on.