Richmond trailed 3-0 until Barry scored in the 48th minute on an assist from Chelsea Davies, but the Spiders still trailed by two goals until the final 10 minutes of the game.
The Spiders then scrambled for the game-tying goal for the final nine minutes, before being awarded a penalty corner with three seconds remaining.
After the final horn sounded, but before Columbia could thwart the Spiders, junior Christina Amorose was able to find the back of the net to force the extra session.
Then Barry made quick work in the first overtime, taking the ball down the left side and beating the keeper to put the game to rest.
Columbia came out unbelievably efficient, scoring on all three of its first-half shots, the first coming just 2:22 into the match. The Lions made it 2-0 8:49 and 3-0 in the 35th minute.
At the half, Richmond had out-shot Columbia 15-3, but trailed 3-0. The Spiders held a 30-5 advantage in shots for the game.
Richmond hosts Georgetown on Sunday at 1 p.m.