The Cumbrians hoped to extend their three-game unbeaten run, while second-placed MK Dons were looking to make it 11 wins from 13 starts. United made a bright start when they won a corner and after a scrambled clearance, Dobie rattled the post.
But it was MK Dons who opened the scoring. Aaron Wilbraham had threatened with a tenth minute header just wide of the post, but he went one better on the quarter-hour after collecting Jemal Johnson's cross to fire past Tim Krul.
United had a chance to equalise in the 27th minute when Graham Kavanagh's free-kick found Dobie, but he couldn't steer his effort on goal and United were to rue that miss before half-time. A groin injury to Jeff Smith saw him replaced by Simon Hackney and as United reshuffled, only a Danny Livesey goal-line clearance prevented a second for the Dons.
But the visitors doubled their tally when Wilbraham got his second, slotting the ball past Krul from just inside the area despite furious protests from the goalkeeper. The decision incensed United and in a fractious few moments, Danny Graham and Michael Liddle were booked.
United exploded after the restart and pulled a goal back within seconds. Graham turned Miguel Angel Llera to pick his spot with an angled shot inside the box. Man-of-the-match Paul Thirlwell carried the ball from midfield in the 56th minute only to fire in a tame shot, but he levelled in the 65th with a piledriver from just inside the box.
United now had their visitors on the back foot and it was no surprise when Anyinsah fired the Cumbrians in front, but it was drama to the end. Wilbraham wasted a great chance to put the Dons level but he somehow put the ball over the bar from inside the six-yard box after Krul had blocked Johnson's shot.
In the final moment of the four minutes of time added on, Graham was upended by Dons keeper Willy Gueret, but the referee blew for full-time rather than give a penalty. The win for Carlisle dragging them nine points clear of the relegation zone while MK Dons fell to their first defeat on the road since mid-September.