It leaves Antrim in the relegation play-off against Carlow, after picking up just two points in the campaign.
Referee Alan Kelly left the Belfast venue with the jeers of the home support ringing in his ears and afterwards Saffrons boss Kevin Ryan questioned a number of decisions made by the Galway official.
“I never talk about referees but I just can’t believe that (performance). It seemed that any time we were going to get a lead the referee was going to stop us anyway,” said Ryan.
The sides were level seven times during the opening half and were still deadlocked, 0-11 each, at half-time.
On the 46th minute they were level for the eighth time when Ciaran Slevin landed his third from play, but four unanswered scores from the home side looked like securing a second victory of the campaign.
After 57 minutes the gap remained at four points when Tony McCloskey and Slevin swapped scores, but the Saffrons failed to raise another flag and Offaly shot six points, with Dooley converting an injury-time free to see the visitors over the line.
ANTRIM: R McGarry, C McKinley, A Graffin, M McFadden; S McCrory (0-1), N McAuley, P Shiels (0-2); M Donnelly, J McGreevy; S Beatty, J McManus (0-12, 10f), E McCloskey (0-1); D McClean, C Carson, S McNaughton (0-1).
Subs: T McCloskey (0-1) for M Donnelly, S McAfee (0-1) for D McClean.
OFFALY: J Dempsey; C McDonald, D Kenny, P Cleary; P Camon, R Hannify (0-1), K Brady (0-1); C Slevin (0-7, 4f), B Carroll; C Egan, D Molloy, G Healion; S Dooley (0-10, 7f, 0-1 sideline), J Mulrooney (0-1), T Carroll.
Subs: C Pardon for T Carroll, O Kealey for D Molloy, S Ryan for P Camon.
Referee: A Kelly (Galway)