Chadwick netted two close-range finishes before David Goodwillie nodded in an 83rd minute consolation goal for Rovers as the Dons claimed a richly-deserved giant-killing win.
The home side controlled the first half and Blackburn goalkeeper Jake Kean produced good saves to deny Daniel Powell and Chadwick, while Charlie MacDonald clipped an effort just wide.
Rovers, showing seven changes from the weekend, rarely threatened before the break but Edinho Junior missed their best chance when his low shot was cleared off the goal-line by Shaun Williams.
The Dons burst out of the blocks after the interval and Chadwick calmly slotted home Jon Otsemobor's cutback to put them ahead after 51 minutes.
Chadwick doubled the Dons' lead 15 minutes later when he tapped in the rebound after Powell's shot had been spilled by Kean, before Goodwillie headed in Fabio Nunes' cross late on for Rovers' consolation goal.