Platt's treble took his season's tally to 11 as the Dons were made to work hard for the points by the gritty Bees, who had Dean Sinclair and Ian Hendon sent off in the first half after Simon King's equaliser.
Scott Taylor wasted a golden opportunity to put the Dons in front after just 40 seconds when he raced onto Dean Lewington's through ball but was denied one-on-one by Barnet goalkeeper Lee Harrison from a tight angle.
Yet the Dons didn't have to wait long to take the lead and Platt broke the deadlock with a cool one-on-one finish in the 12th minute. Platt intercepted Nicky Bailey's mis-cued back-pass to race through on goal before drilling a low strike past Harrison into the corner of the net from 15 yards.
But Barnet battled back and just minutes after he had fired a good chance wide, King went one better by equalising for the Bees in the 27th minute. Bailey's free-kick was flicked on by Ismail Yakubu and King stole in unmarked to plant a well-directed into the corner from six yards.
Yet Barnet's joy turned to despair a minute later when Sinclair was shown a straight red card for a dangerous two-footed tackle on Dons midfielder Aaron Wilbraham on the halfway line.
And then the Bees were reduced to nine men on the stroke of half-time when skipper Hendon was given his marching orders for landing an elbow in Lewington's face as the pair contested a high ball near the touchline.
Dons piled forward in the early stages of the second half and Sean O'Hanlon hooked a seven-yard half-volley just over the bar before Jude Stirling belted a thunderous left-foot drive inches wide.
But the nine-man Bees showed plenty of spirit and Jason Puncheon missed a great chance to score against his old club on the counter-attack when he volleyed Adam Birchall's deep cross over the bar from eight yards in the 56th minute.
And Platt made him pay when he grabbed his second goal of the game to put the Dons 2-1 up in the 65th minute. Jon-Paul McGovern's deep free-kick was headed back into the danger zone by Wilbraham and Platt controlled the ball before hooking it over his shoulder and into the roof of the net from five yards.
Platt completed his hat-trick in the 70th minute when he rammed in the rebound after his penalty had been saved by substitute keeper Ross Flitney, following Bailey's push on Dons substitute Jonathan Hayes in the box.