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Callum Maycock put the hosts in front two minutes into the second half, but substitute Evan Mooney struck back 25 minutes later to set up a frantic finish.
With both sides keen to win the tie without the need for penalties, Emerson Nwaneri, who had set up the equaliser, netted the winner from close range in the 87th-minute, taking us through at Wimbledon's expense.
Press play on the video at the top of the article to watch highlights from our comeback victory.