David Partridge's 88th minute own goal looked to have won it for Rovers, but substitute Izale McLeod equalised for the Dons in the 90th minute before Facey unbelievably struck the winner two minutes later.
Dons manager Danny Wilson kept the same starting line-up that won 2-1 at Chesterfield last Saturday, with Wade Small returning from injury to take his place on the bench.
Tranmere manager Brian Little was dealt a blow before kick-off when Ian Goodison got injured in the warm up, so Carl Tremarco took his place.
Milton Keynes started brightly and the lively Aaron Wilbraham raced onto Clive Platt's flick but his left-foot strike was deflected just wide by Paul Linwood, before he forced Rovers goalkeeper Steve Wilson into a low save from the edge of the box in the 18th minute.
But Tranmere skipper Michael Jackson almost gave the visitors the lead against the run of play in the 26th minute, when his stooping close-range header from Jason McAteer's corner flew inches over the bar.
Rovers keeper Wilson had to be alert again to hold Gary Smith's stinging 22-yard drive at full stretch just after the half-hour mark.
At the other end, Delroy Facey forced Dons keeper Matt Baker into his first save of the afternoon with a rasping 30-yard strike.
But the Dons continued to dominate and Platt was denied by a terrific last-ditch tackle from Ian Sharps in the box just before the break.
The diving Platt was within a whisker of converting Smith's dangerous inswinging cross at the near post five minutes after the break but the ball flashed past his head.
And then Chris Greenacre planted a free header wide for Rovers from 12 yards from Linwood's cross just moments later.
McAteer curled a dipping 25-yard free-kick onto the top of the bar with Baker well beaten as the Dons rode their luck, before Greenacre headed McAteer's cross straight at Baker from close range in the 74th minute.
Partridge headed McAteer's free-kick into his own net in the 88th minute to give Tranmere a late lead and spark a tense game into life, but McLeod looked to have earned the Dons a point when the ball dropped kindly for him in the box and he planted a spectacular overhead kick into the corner in the 90th minute.
Yet Facey capitalised on a defensive slip by Partridge to race through on goal and slot the winner under Baker two minutes into injury time.