Platt's 57th minute header keeps the Dons six points behind third-placed Swindon but ended Darlington's outside hopes of reaching the play-offs.
The Dons totally dominated the first half and Gary Smith's 25-yard drive fizzed just wide before Dean Lewington slashed an angled half-volley inches past the top corner from 15 yards after a neat one-two with Leon Knight in the ninth minute.
Milton Keynes' winger Jon-Paul McGovern's 20-yard curler was deflected just wide before the stretching Knight headed Lloyd Dyer's well-flighted cross straight at Darlington keeper Sam Russell from 10 yards on the quarter of an hour mark.
And then McGovern's acrobatic overhead kick from Platt's header flew straight at Russell from 12 yards before Dyer's curling 25-yard free-kick shaved the post three minutes before the break.
All the defensive-minded Quakers could muster in response in the first half was a long-range strike from Julian Joachim that he dragged well wide.
Darlington adopted a more attacking approach after the break and Marcus Phillips headed Rory Prendergast's in-swinging corner just over the bar.
The only goal came in the 57th minute from Platt, when the 6ft 4in striker nipped in ahead of his marker to head Knight's pin-point cross past Russell from eight yards for his 17th goal of the season.
The Quakers battled back and Don's keeper Ademola Bankole spilled skipper Mickey Cummins' fizzing long-range free-kick and blocked Gregg Blundell's 10-yard effort before the unmarked Patrick Collins headed straight at the keeper from five yards in the 75th minute.
Joachim then twice went close to equalising when first he turned Prendergast's low cross against the bar in the 80th minute, before he raced clean through a minute later but his powerful shot was well saved by the diving Bankole.