Disappointment for Banbridge as after being ahead for the whole game they conceded a last kick of the tie penalty to give Antrim Grammar the narrowest of victories.
Banbridge coach Brian Leslie said: "It was a tough game and I know we have not had the best of seasons with injuries but this was a game that should have been closed out sooner.
"We took a deserved 10-3 interval advantage but mistakes were made at key times and we had a lot of possession in the middle but didn't do a lot with it. Carrick gained confidence and came back at us. I was really gutted at the end as I and all the home spectators thought we had won.
"In injury time we dispossessed Carrick in midfield and there was a large cheer as we waited for the final whistle but instead the referee took play back to in front of our posts and gave Carrick a penalty. What an anti-climax."
Banbridge were in front after seven minutes when a driving maul led to a try by prop John Wilson but the visitors replied with a Josh Battishill penalty. On the half-hour home flanker Jonny Campbell picked up from five metres out to dive over. The trend continue after the restart when scrum-half Jordan Mount picked up and dashed in from 40 metres out after five minutes. Unfortunately for them none of the three tries was goaled.
Antrim had a breakaway try by winger Adam Johnston converted by Battishill and they were back in the contest. On the half-hour Sam Gordon, who had replaced Battishill, kicked a penalty and he later sealed it with his second.
Robin McKenna, the Antrim coach was thrilled. "They defeated us 45-0 in the first round last season so it was pleasing to see how much we have improved. I thought JJ McKee and Adam Johnston had brilliant performances. It was an amazing result and it's the first time Antrim has reached the second round."