It was no more than League One side Colchester deserved for a dominant display and it was a match that they could have won in normal time.
Premiership outfit West Brom were second best for virtually the whole night apart from a brief period in the second half when Geoff Horsfield drilled home to equalise out of the blue.
But the Baggies could not get themselves out of jail and May, on loan from Millwall, came up trumps with the winner in the 118th minute.
Australian Richard Garcia rose high to meet Joe Keith's corner with a header that was blocked on the line by West Brom midfielder Lloyd Dyer.
But striker May lunged in bravely to stab home the loose ball from close range to net his second goal for the Essex club.
The Layer Road crowd went wild and Colchester held on for the last three minutes of extra time to record an historic victory.
Earlier, Craig Fagan had broken the deadlock with a marvelous finish in the 29th minute.
It was a great move involving Gareth Williams and Fagan had time to turn and send a rising shot into the roof of the net from just inside the box.
It was Fagan's sixth goal of the season and it was extra special because earlier in his career he had turned down a chance to sign with West Brom. He chose deadly rivals Birmingham instead.
United could have added to their tally before the break and Garcia should have scored a second goal when blazing over with an empty net beckoning after keeper Tomasz Kuszczak could only parry a shot from Neil Danns.
Against the run of play West Brom grabbed the equaliser when Horsfield sprung the Colchester offside trap to run on to Thomas Gaardsoe's pass and beat keeper Aiden Davison.
There was more than a hint of offside but the assistant referee kept his flag down.
Colchester came back strongly and were the beater team again in extra time.
Keith peppered the target with three shots but in the end it was May, on as a second half substitute, who came up with the winner.