The visitors went ahead in the 22nd minute when Wimbledon goalkeeper Seb Brown fluffed Daniel Kearns' cross and pushed the ball into his own net.
The Dons were level 12 minutes later when Strutton bent the ball into the net following a pass from Rashid Yussuff, who was subsequently felled in a high, rash challenge from Scott Kerr.
It was Kerr's second yellow, condemning York to play with 10 men for 55 minutes.
With 13 minutes to go Strutton scored his second, getting in front of a hesitant York defence and firing home after Michael Ingham had parried his initial shot.
York equalised in the last minute, Jamie Reed taking the ball down with his right foot before coolly slotting home a minute after coming on as a substitute.
Two goals in two minutes put the hosts firmly in control in extra-time.
The first, seven minutes into the extra 30, saw York defender Clarke Carlisle misjudge Warren Cummings' cross from the left and the ball ricocheted in off Byron Harrison.
Stretton then turned provider, heading back across goal to Jack Midson in the six-yard box and he had a simple job of turning the ball across the line.
Reed bundled the ball home with two minutes to go but Wimbledon held on to book a second-round tie with MK Dons should the League One side beat Cambridge City on Tuesday.