Ben Hamer made a smart reaction save to keep out Alan Smith's 36th-minute header, but it was Charlton who broke the deadlock two minutes before half-time in controversial circumstances.
Chris Solly's powerful drive was hacked away by the visiting defence, but as the Dons tried to clear their lines, there was a clash between Yann Kermorgant and Gary MacKenzie.
MacKenzie appeared to push his head forwards towards the Charlton striker's face and the offence earned him a straight red card, as well as gifting the home side a penalty, which was coolly despatched by Jackson.
And it was Jackson who won the Addicks' second spot kick in first-half stoppage time after he was sent to ground just inside the area by Luke Chadwick. The Charlton captain was quickly on his feet and this time smashed his powerful shot into the bottom right corner to put the hosts in command at the break.
Bradley Wright-Phillips should have added a third midway through the second half but he fired wildly off target from 15 yards before Dean Bowditch pulled one back in the 87th minute to set up a tense finish.
