Platt rose to head home a Mark Yeates cross with seven minutes left as Millwall's struggle at home continued.
The Lions looked by far the most threatening side in a goalless first half with Gary Alexander a constant threat until he went off injured just before the break with a head injury.
Alexander came closest to opening the scoring after 15 minutes but his header from David Martin's cross was pushed round the post by Mark Cousins.
Four minutes after the break Martin shot tamely at Cousins before David Perkins stung the palms of Millwall keeper David Forde at the other end. Colchester offered very little else going forward and the game looked to be heading for a draw until Platt clinched all three points late on.
The result means Millwall have won just twice at home since Christmas to offset a club-record six straight wins on the road.