The Milton Keynes side had a couple of decent chances, hitting the woodwork with one, but Posh dominated and should have run up a cricket score - as it was they were happy to settle for goals from Grant McCann and the excellent Craig Mackail-Smith.
Posh will now travel to Old Trafford, home of Ferguson’s father Sir Alex, where they face Lee Clark’s Terriers in the May 29 final.
Chasing a 3-2 first-leg deficit, Posh started ferociously as the visitors barely left their half.
David Martin just about kept Mackail-Smith’s low drive out at his near post before, 10 minutes in, McCann opened the scoring with a cheeky dipping free-kick that the MK Dons goalkeeper should have stopped at his near post, although he was not helped by a cowardly effort from the wall.
The visitors rallied well in the following minutes but the next big chance fell to Peterborough, as Lee Tomlin’s vicious low ball across goal was missed by the sliding Mackail-Smith and James Wesolowski.
Posh were creating more - Mackail-Smith again testing Martin at his near post - but the best chance of the half bar the goal dropped to MK Dons as Daniel Powell thrashed a shot against the crossbar only to see it bounce a yard short of the line.
The second half started relatively scrappily, the first incident of note a late tackle from Luke Chadwick that earned the former Manchester United trainee a booking.
But eight minutes in it was a product of non-league that had the next say as, after George Boyd forced a good stop from Martin, Mackail-Smith prodded home his 34th goal of the season after his initial effort was blocked on the line by Chadwick.
As with the first goal, MK Dons responded well as Angelo Balanta caused problems from the left, but Posh should have been three up seconds later but for a poor piece of refereeing by Colin Webster.
Tomlin smacked a superb low drive inside the far post but play was pulled back as Webster had decided that, in the movement that saw Boyd release his team-mate, Stephen Gleeson had committed a foul.
He had done no such thing - Boyd either slipped or dived - and, even if he had, the advantage was clear as Tomlin was sent into the box.
McCann’s resultant free-kick was blocked and the error was almost compounded with a Dons goal as QPR loanee Balanta raced clean through down the other end: he was foiled by a marvellous tackle from Wesolowski.
With 20 minutes remaining visiting defender Gary MacKenzie prevented another Posh goal, clearing Tommy Rowe’s effort off the line after great play by Tomlin and Boyd to create the chance.
The last quarter-hour was a scrappy affair, punctuated by bookings and with a strange flashpoint as MK Dons defenders Dean Lewington and Keanu Marsh-Brown almost came to blows.
Lewington - who had just been booked for shoving an opponent - almost scored a spectacular headed own goal, his embarrassment avoided by an alert Martin, while down the other end blushes were spared for Balanta who missed a gaping goal with a howler of a finish only to be ruled offside anyway.
The closing minutes - including a whopping five of injury time - were all Peterborough who looked more like a side desperate for a goal than one clinging on to a narrow aggregate lead: Mackail-Smith hit the post, Chris Whelpdale fired inches wide and Tommy Rowe shot straight at Martin when put clean through by Mackail-Smith.
So dominant were Posh that Dons barely got a sniff late on as the hosts comfortably closed out a win that puts them in a final against Huddersfield in 10 days: the omens are good for Peterborough, who have won both play-off finals they have contested.