Scunthorpe, who came into the match on the back of six straight League wins, looked to have the points won at half-time after Izzy Iriekpen and Paul Hayes had put them in the driving seat.
The turning point came in the 55th minute when Scunthorpe keeper Joe Murphy saw red for bringing down John Melligan - and that sparked Orient's revival.
Scunthorpe looked dangerous at set-plays and took the lead on 24 minutes following Grant McCann's corner from the right.
Orient defender Alton Thelwell could only head the ball down into the path of Scunthorpe skipper Izzy Iriekpen who hammered the ball home.
The visitors looked to have wrapped things up in first-half stoppage time when McCann whipped in a free-kick and Hayes, unmarked, headed home from eight yards.
Two minutes into the second half Hayes had a penalty claim turned down by referee Anthony Taylor, and Orient quickly cashed in on their good fortune.
With ten minutes of the second half played, Melligan raced clear of the visiting defence and was brought down by Murphy as he rounded the keeper. Referee Taylor pointed to the spot and pulled out a red card.
Scunthorpe sacrificed Martyn Woolford and threw sub keeper Josh Lillis into the action. His first touch saw him palm away Boyd's spot-kick but the Orient striker followed up to smash the ball home.
The goal fired up the home side and on 67 minutes skipper Purches popped up in the penalty area and latched on to Boyd's pass to rifle home his first goal of the season.
Simon Dawkins almost won the game for Orient but was denied by Lillis, while at the other end Glenn Morris made a great, late stop to deny the dangerous Gary Hooper.