Darren Ferguson's side were gifted two first-half goals to take control of the two-legged tie, with an error from Wednesday keeper Cameron Dawson and a deflected shot from Joe Ward leaving the visitors playing catch-up.
Things only got worse in the second half too, as Kwame Poku and top-scorer Jonson Clarke-Harris arguably put the tie beyond doubt at the halfway stage, with Peterborough already having one foot in the Wembley final.
Posh had only snuck into the playoffs by virtue of results going their way on the final day, whereas Wednesday were in the automatic promotion places from January to April and finished the regular season in third with a whopping 96 points - a tally high enough to have earned automatic promotion in every previous League One season, and which would have won them the title last term.
Despite 19 points separating the two sides in the final table, underdogs Peterborough ran riot in front of their delirious home support, although they were given a helping hand in the first half.
Jack Taylor broke the deadlock after 20 minutes when his strike from the edge of the box squirmed through the grasp of Dawson, who should have done better in the Wednesday goal.
Indeed, the general play was fairly even between the two sides for much of the first half, yet Darren Moore's men found themselves two down nine minutes before the interval when Ward's long-range strike deflected off the back of Marvin Johnson, who was slow to close his man down, and looped past the keeper.
Wednesday had chances of their own to get themselves back into the match, but they were wasteful whereas Peterborough were clinical at the other end.
The hosts added a third goal five minutes into the second half when Ephron Mason-Clark's cross was finished off by Poku, and any hope of a spectacular second-leg comeback became even bleaker when Clarke-Harris added a fourth late on.
Dawson initially appeared to have gone some way towards making up for his earlier error with a sensational save to deny Ricky Jade-Jones, but Peterborough kept the ball alive and fed it back in to Clarke-Harris, who nodded home his 27th goal of the season.
The result leaves Sheffield Wednesday needing a four-goal victory when they welcome Peterborough back to Hillsborough for the second leg next Thursday, or else Posh will book a place at Wembley to take on either Barnsley or Bolton Wanderers.