Platt's double and Izale McLeod's super solo goal fired the Dons to their first win in four games, after Bradley Thomas and Drewe Broughton had put lowly Boston 2-1 up at half-time.
Platt gave the Dons a dream start when he powerfully headed home his 12th goal of the season to put them in front in only the third minute.
Skipper Keith Andrews collected Jamie Smith's quick throw before delivering a searching ball into the box, and Platt out-jumped his marker Jamie Stevens to thump an unstoppable header into the top corner from 12 yards.
McLeod almost made it 2-0 just a minute later but Boston goalkeeper Andy Marriott saved his close range strike superbly from Smith's cut-back.
But the Pilgrims battled back and ex-Don Albert Jarrett's 25-yard drive was well saved by Milton Keynes keeper Adolfo Baines, before Broughton headed over at the near post.
McLeod and Jon-Paul McGovern were just off target for the Dons with rasping long range drives, before Thomas equalised for Boston against the run of play in the 21st minute. He nipped in ahead of Dons keeper Baines' attempted punch at the near post to head Jarrett's pacey corner into the net from six yards.
The Dons came roaring back and Lloyd Dyer's 12-yard strike was deflected inches wide before Paul Butler's close range header from McGovern's corner thumped against Marriott's crossbar in the 22nd minute.
But Boston weathered the storm and Broughton caught Milton Keynes with another suck-punch at the other end to put the Pilgrims 2-1 up in the 32nd minute. Colin Cryan's free-kick was flicked on by Nathan Joynes and Broughton swivelled 12 yards out to blast an emphatic strike past the rooted Baines and into the top corner.
Broughton had another effort ruled out for offside in the 38th minute, before Sean O'Hanlon had a header cleared off the line by Boston centre-half Thomas at the other end on the stroke of half-time as the Dons rallied.
Milton Keynes dominated after the break and McLeod was denied what looked like a blatant penalty in the 55th minute when he was pole-axed by Stevens after nipping the ball away from the Pilgrims' centre-half in the box.
O'Hanlon's 10-yard strike from substitute Gary Smith's corner was cleared off the line by Pilgrims' wing-back David Farrell, before McLeod equalised for the Dons with a fantastic individual goal in the 75th minute.
McLeod cut in from the left flank and skinned Boston centre-half Cryan before curling a superb 12-yard finish inside the far post from an acute angle.
And Platt completed his brace to win it for the Dons six minutes from the end when he hooked Aaron Wilbraham's knock-down low into the corner of the net from seven yards, extending the Pilgrims' winless run to nine games.