Mansfield had gone into the clash on the back of three successive clean sheets at home, but were tamely swept aside as Peterborough created a host of chances.
John Mullins did head against the bar from a Matt Hamshaw corner in the ninth minute for the home side, but after that it was all one-way traffic as the Posh's slick play proved too good for the disappointing Stags.
Muggleton denied Shane Huke, Adam Newton, Crow and Richard Butcher with some fine stops before McLean breached the Stags' rearguard in the 39th minute - the first home goal Mansfield had conceded in 313 minutes.
The lively striker notched his third goal in four games since his move from Conference side Grays by powering home a bullet header from skipper Dean Holden's corner.
It was no more than the visitors deserved and on the stroke of half-time they nearly grabbed a second following a quick breakaway involving the lively McLean, who found Newton unmarked in the box - but the winger saw his fierce drive saved by the in-form Muggleton.
The visitors began the second half as they ended the first by laying siege to Muggleton's goal. The Stags' keeper was forced to tip over a hanging Stirling cross in the 46th minute and from the resulting corner saved a close-range Guy Branston header.
Stags' first real opening of the second half came in the 62nd minute when Hamshaw played an inch perfect ball through to Richie Barker, whose fierce shot from the right was tipped away by Tyler.
But in the 67th minute the home side's hopes of getting back into the game were well and truly dashed when Crow made it 2-0 with a close range effort. He timed his run to perfection to meet Branston's flick on from Stirling's long throw from the right.
Mansfield boss Peter Shirtliff threw on livewire strikers Danny Reet and Michael Boulding in a bid to force a way back into the game, but they had little impact as the Posh rearguard held firm.
In the 88th minute Coke almost notched up a consolation for the home side, but headed over the bar from a pinpoint Simon Brown cross -while at the other end substitute Lloyd Opara fashioned two late efforts.