Posh had not won on December 26 since 1996 - when the Lions were also their opponents - but Mclean netted the only goal with a well-placed 23rd-minute shot after being released by strike partner Craig Mackail-Smith.
But the home side were unable to turn a numerical advantage into further goals after Millwall defender Tony Craig was dismissed for a deliberate elbow on Chris Whelpdale in the 32nd minute.
Mackail-Smith saw an effort ruled out for offside soon after and then fired another good chance into the side netting after the break.
Wideman George Boyd was also close when blasting straight at Millwall keeper David Forde.
Striker Gary Alexander could only steer a free header wide 12 minutes from time when presented with the opportunity to pinch Millwall an unlikely equaliser.