In the first half Offaly struggled and a very honest Antrim side fully deserved their 0-9 to 0-5 interval lead.
Offaly were much better in the second half with Joe Bergin registering nine points in all, seven from frees.
The decisive score came 21 minutes into the second half when Conor McGinley came under pressure from Sean Gardner and hand passed the ball into his own net. That gave Offaly a 1-13 to 0-12 lead and they powered home from there.