The League Two outfit have never played the formerly named Wimbledon F.C since they relocated to Milton Keynes in 2004, but Byron Harrison and Jack Midson goals in the additional half hour saw off 10-man York and set up the potentially historic fixture.
Goalkeeper Brown had gifted City a 22nd-minute lead when he punched the ball into his own net under the challenge of York defender and PFA Chairman Clarke Carlisle.
Strutton equalised for the hosts with a shot from the edge of the area in the same incident that saw York reduced to 10 men.
Minstermen midfielder Scott Kerr was shown a second yellow card for a foul in the build-up to Strutton's goal, forcing the visitors to play the remainder of the game with a man less.
They resisted Wimbledon attacks for the majority of the second half but were finally broken down when Strutton doubled his tally 13 minutes from time.
However, with his first kick since being introduced as a late substitute, Reed levelled the scores.
That goal was only to be a temporary reprieve for York as Harrison and Midson struck in the space of two minutes to give Wimbledon an two-goal cushion that they would not surrender, despite another Reed effort minutes prior to the final whistle.
Should the Dons overcome Cambridge City in their own first round replay, they will play Wimbledon on the weekend of December 1 and 2.