First to go for the home side was substitute Byrne for his challenge on winger Chris Taylor in the 61st minute. Three minutes later United were reeling again when Cleveland Taylor got his marching orders for a second bookable offence.
Earlier Oldham had dominated the first half without being able to find their finishing touch in front of goal.
Skipper Will Haining headed a free kick straight at goalkeeper Joe Murphy while midfield man Gary McDonald fired in a 20-yard shot that was deflected just wide.
The visitors were unlucky in the 16th minute when striker Paul Warne found the back of the net with a close range header, only to be flagged offside.
Scunthorpe struggled to get into their stride in the opening 45 minutes, but suddenly found themselves in front nine minutes into the second half.
It came when Ian Morris delivered a curling cross from the left which was headed into his own net by Oldham centre-back Sean Gregan.
The home side didn't have the chance to build on that lead before falling foul of referee Darren Deadman, who also booked nine players to add to his two red cards.
For the final half-hour Scunthorpe were forced to defend desperately in and around their box.
Goalkeeper Murphy was on top form, keeping out close range shots from leading marksman Chris Porter and from winger Chris Taylor.
It was four minutes into stoppage time when Oldham finally salvaged a point with midfield man McDonald popping up to rifle the ball home from six yards after a real goalmouth scramble.
It was an equaliser the visitors deserved but it was bad luck on Scunthorpe who had defended so heroically for so long.