His ninth goal of the season ensured Darren Ferguson's men delivered the club's first December 26 victory since 1996, when Millwall were also the opposition.
McLean struck in the 23rd minute when collecting a flick from strike partner Craig Mackail-Smith and finishing smartly into the far corner with an effort that gave Lions keeper David Forde no chance.
Posh's grip on the game tightened nine minutes later when Millwall defender Tony Craig was dismissed for a deliberate elbow on Chris Whelpdale, but the home side couldn't convert their numerical advantage into further goals.
Mackail-Smith thought he had doubled the advantage when slotting home at the end of a lightning 35th minute breakaway, but an assistant referee's flag cut short the Posh celebrations.
And from then on it became a frustrating early afternoon for Ferguson's aces. Mackail-Smith drilled into the sidenetting early in the second period, before wideman George Boyd steered an effort against an upright.
Boyd forced a fine save from Forde with another chance as Posh repeatedly threatened without joy - something which ensured a nervy climax.
Lions striker Gary Alexander failed to make the most of a golden opportunity to grab his side an unlikely equaliser when putting a free header wide 12 minutes from time, but the alarm bells continued to ring.
Milwall substitute Neil Harris saw a fiercely-struck volley diverted away from danger by Charlie Lee's head, before fellow replacement David Martin tricked his way past a couple of defenders in stoppage time only to send his shot straight at keeper Joe Lewis.