Visiting keeper David Martin was left red-faced as he slipped on the rain-sodden pitch and saw the ball bounce past him, gifting Kuqi the easiest of break-throughs.
MK Dons went close on six minutes when Luke Chadwick was denied by keeper Alex Cisak but Oldham took control with a string of chances. Tom Adeyemi and Filipe Morais fired off target and left-back David Mellor was unlucky not to score at the end of a storming run.
Oldham struck on 43 minutes as Martin came out to gather Zander Diamond’s long ball but lost his footing to give Kuqi a tap-in from two yards.
Kuqi should have scored again before half time but it was Adeyemi who doubled the lead on 56 minutes as he took Robbie Simpson’s pass and slotted into the far corner from 10 yards.
Chris Taylor was denied by a goal-line clearance and Morais drilled inches wide as Oldham continued to impress, but Dons responded five minutes from time when Clinton Morrison slid in to convert Anthony McNamee’s left-wing cross.