Platt pounced with an 85th minute header after Dons skipper Keith Andrews' penalty had cancelled out Lee McEvilly's early opener for the lowly Dragons, who had Lee Roche sent off in the first-half.
McEvilly put Wrexham ahead against the run of play in the 12th minute, when skipper Neil Roberts beat the offside trap and crossed unselfishly for the unmarked Northern Ireland international striker to slide in his seventh goal of the season from close range.
Andrews equalised for the Dons in the 38th minute when he coolly sent Dragons' keeper Mike Ingham the wrong way from the spot for his third goal of the season, after Roche was shown a second yellow card for handling Andrews' shot in the box.
Milton Keynes upped the tempo after the break and Izale McLeod had a goal ruled out for offside in the 47th minute after Jon-Paul McGovern's 22-yard chip had been tipped onto the crossbar by Ingham.
Dean Lewington volleyed McGovern's well-worked corner inches wide from 20 yards before Gary Smith's 22-yard curler hit the post in the 66th minute as the Dons' frustrations grew.
Dons substitute Leon Knight then turned sharply in the box but his low 16-yard drive from Platt's knock-down fizzed just wide of the post.
But Platt popped up five minutes from the end to head substitute Jon Hayes' chipped cross into the corner of the net from 10 yards and break Wrexham's hearts.
Substitute Jude Stirling and McLeod fired just wide for the Dons after the Dragons were reduced to nine men when winger Simon Spender replaced the crocked Ingham in goal for the dying minutes.