It was a rewarding result for new Tranmere boss Ronnie Moore who was dismissed in the close season by Oldham but he did have some sympathy for the sending off decisions.
Substitute Chris Taylor was red-carded in the 88th minute for bringing down the substitute Calvin Zola who was judged to have been through on goal.
Teenager Neal Eardley followed in injury time for kicking the ball at Zola who had been tripped by Chris Swailes.
There was never much between the sides. The only goal came in the 25th minute. Greenacre headed into the top right corner after Kevin Ellison headed on a cross by Chris Shuker.
Tranmere included nine new signings with only Greenacre and Ian Goodison from last season's squad starting the match.
It needed an excellent diving save by Latics keeper Les Pogliacomi to deny a drive by Ellison in the 28th minute.
The Merseysiders should have doubled their advantage four minutes into the second period but Greenacre this time missed a sitter, failing to find the target from 10 yards when set up by Paul McLaren.
Greenacre was also out of luck when an angled shot shaved an upright in the 77th minute.
Neil Wood was outstanding in midfield for Oldham and there were also some sharp touches from Paul Edwards.
But loan signing Maheta Molango had no luck with two first-half efforts and in the 57th minute brought a near post save from home keeper Gavin Ward with a powerful header which followed an Edwards centre.
Oldham never threw in the towel and three substitutes joined the action in the closing minutes. Taylor went nearest to an equaliser with a header from the edge of the box which fell a yard outside an upright.
Shuker was a big influence up the right-wing for Tranmere and Ian Goodison and Chris McCready remained resolute in central defence.