O'Connor, who had spurned several chances beforehand, finally equalised in the closing stages when he bundled in a close-range shot.
Hibs lost midfielder Scott Brown late in the first half for two yellow cards but came back strongly and skipper Gary Caldwell was just inches too high with a header in the final seconds, with Dunfermline on the ropes.
The home side started brightly and Michael Stewart set up O'Connor, but he pulled his shot well wide of target.
Dunfermline hit back and Zbigniew Malkowski did well to hold a low shot from Scott Morrison after Simon Donnelly had cut the ball back.
But Hibs did not heed the warning and Jim Leishman's side snatched the lead after 12 minutes, when Darren Young set up Greg Shields and he cracked home a low shot from 16 yards that left Malkowski with no chance.
O'Connor had another opportunity after a Steven Fletcher shot was blocked but he could get no power in his effort and Bryn Halliwell nabbed the loose ball.
Derek Riordan went close ten minutes from the break with a 30-yard free-kick which dipped just over the top.
O'Connor then cracked a shot against the post after 39 minutes after Stewart sent him clear, but Hibs were reduced to ten men a minute later.
Scott Brown, booked midway through the half, fouled Darren Young and referee Charlie Richmond was left with no option.
O'Connor missed another chance for Hibs after a well-worked free-kick when he steered his shot wide.
The 22-year-old finally got on the scoresheet after 74 minutes when Halliwell saved a Riordan shot and he was able to bundle the ball home.
Halliwell then saved well from Gary Smith as Hibs went for the winner, but Dunfermline just managed to hold on.