The 6ft 4in striker sent Lincoln crashing out with two instinctive finishes either side of Paul Mayo's spot-kick.
Dons boss Danny Wilson made two enforced changes from the team that won 2-1 at Tranmere on Saturday, with Nicky Rizzo and Gareth Edds coming into the midfield for on-loan duo Stephen Quinn and Pablo Mills, who were denied permission to play by their clubs.
The visitors, struggling in 16th place in League Two, started brightly and striker Gary Birch should have converted Lee Beevers' cross from four yards in the second minute, but he mis-hit his effort allowing goalkeeper Matt Baker to block.
And Platt made them pay in the 14th minute when his well-taken goal put the Dons ahead.
Full-back Trent McClenahan whipped a left-footed cross over to the back post and the towering striker controlled the ball on his chest before clipping it past onrushing keeper Alan Marriott into the corner of the net from eight yards.
But Mayo equalised from the penalty spot for Lincoln nine minutes later after Craig Morgan pulled down Marvin Robinson as he looked to latch on to Beevers' cross.
Mayo stepped up to drive an excellent spot-kick low to Baker's left as the keeper dived at full stretch without success.
Yet the Dons fought back and Platt grabbed his fifth goal of the season to put them 2-1 up three minutes before the break.
McClenahan put in another dangerous in-swinging cross and Platt nipped in ahead of Colin Cryan to loop the ball over Marriott into the far top corner with his knee.
Keith Alexander's Imps came out fired up for the second half and they piled on the pressure in search of another equaliser.
The visitors rained high balls into the box for much of the second period as the Dons struggled to get out of their own half, and Jamie McCombe went close with a header before Mayo's fierce 20-yard drive shaved the post.
Centre-half McCombe forced Baker into a decent save after spinning 12 yards out and firing low.
And substitute Mahetta Molango almost caught out Baker with a cross that was looping into the far top corner before the keeper desperately clawed it over the bar in the 88th minute.
But the nervy Dons hung on to seal their passage to Roots Hall.