Posh boss Darren Ferguson resisted the temptation to make wholesale changes after Saturday's debacle against Macclesfield and his players responded by sweeping into a two-goal inside the first half hour.
Although Jamie Day suffered a third minute hip injury and was replaced by Tom Williams, Posh forced Stanley on to the back foot with Craig Morgan releasing Craig Mackail-Smith, who rounded Ian Dunbavin before slotting home his 14th goal of the season.
Within 60 seconds, Mackail-Smith was inches away from a second and on 27 minutes his shot was only half stopped by Dunbavin to leave Boyd with a simple tap in.
Boyd made it three after controlling a Chris Whelpdale centre and netting with a left-footed volley and McLean blasted his 21st of the season six minutes before the break.
Stanley pulled one back a minute into the second half with skipper Paul Mullin clipping in an Ian Craney centre and, two minutes after the hour mark, Craney delivered again and this time substitute Shaun Whalley bundled the ball into the net.
With the surface becoming increasingly soaked and not suiting Posh's passing game, Stanley continued to make attacking progress and Roscoe D'Sane forced new £400,000 goalkeeper Joe Lewis into a last-ditch save.
Boyd proved the direct route was best, putting Posh back in the driving seat with his third goal on 76 minutes and two minutes later it was six with McLean cleaning up after a Boyd shot was deflected.
McLean completed his treble in the 82nd minute and Mackail-Smith had the last word with number eight in the final minute.