Tranmere looked on course for a vital win in their hunt for automatic promotion until Vernon denied them 19 minutes from time.
The game was played in awful conditions but both sides produced good attacking football.
Oldham have now gone eight matches without losing at home while Rovers slipped seven points behind League One's leading pair Luton and Hull.
The visitors had Jason McAteer's goal ruled out for offside before taking the lead after 37 minutes.
Gareth Roberts crossed from the right and the referee controversially pointed to the spot after deciding that David Eyres used his arm.
Ryan Taylor sent Les Pogliacomi the wrong way to put Rovers ahead, but Oldham were level two minutes later.
Adam Griffin hit a left-foot shot from 25 yards and saw it take a big deflection to wrong-foot the keeper.
Eyres was denied by an excellent Russell Howarth save and Vernon had a shot cleared off the line before Tranmere made it 2-1 seven minutes after the break.
Iain Hume controlled a cross from David Beresford and superbly crashed in his shot off the underside of the bar from ten yards.
Vernon pulled Oldham back level after a good move that ended when he slotted a drive from 16 yards into the bottom corner.
Oldham went close to completing their fightback in stoppage time but Dean Holden headed just off target.