The Dons looked to be heading for their second successive relegation when Tranmere substitute David Beresford cancelled out Edds' early strike midway through the second half.
But his winner six minutes from time ensured Milton Keynes performed a great escape, and with Torquay losing 2-1 at Colchester, sparking wild celebrations and a pitch invasion at the final whistle.
The fired-up Dons got off to a dream start when Edds emphatically fired them in front after six minutes.
Clive Platt's knock down fell nicely in to the path on the on-rushing midfielder on the edge of the box and he rammed home past the dive of Tranmere goalkeeper John Achterberg.
Izale McLeod came close to doubling the lead a minute later but his close-range effort was well blocked by Ian Sharps, before Chris Dagnall's snap-shot at the other end was well held by Matt Baker.
Rovers' best chance of the first half fell to Ian Goodison four minutes before the break but he scuffed his effort straight at Baker from Carl Tremarco's dangerous free-kick.
McLeod was unlucky with a spectacular overhead kick which struck the bar from Trent McClenahan's cross.
But Tranmere fought their way back into the game and it was no surprise when substitute Beresford curled home a 65th minute equaliser.
The midfielder surged down the left and as the defence backed of he cut in and unleashed a sweet 22-yard strike which beat Barker's despairing dive.
Rovers began piling on the pressure and after substitue Eugene Dadi's half-volley hit the bar, Theodore Whitmore and Beresford missed by inches.
Then completely against the run of play, Edds latched on to another Platt flick in the 84th minute and fired home a stunning volley to keep the Dons in the division.