After Chris Porter had notched his ninth goal in eight games, Cox was harshly ordered off for hauling the striker down in the area, allowing Andy Liddell to score from the spot.
And Ben Chorley's second-half own goal added further misery to the Gills as they suffered a rare home defeat.
Both sides came into the game in a festive mood having recorded successive wins over the Christmas period, but struggled to deal with the sodden Priestfield pitch early on.
Oldham's Liddell flashed an early effort over and in the 14th minute saw another shot blocked by Chorley, with Oldham's players claiming the defender used his arm to intervene.
But on 26 minutes Liddell's hopeful lofted pass caught Dean Brill napping and Porter was on hand to loft the ball over the stranded keeper into the net.
And just seven minutes later Gillingham were reduced to ten men when Cox hauled down Porter in the area, denying the striker a goalscoring opportunity according to referee Paul Taylor. Liddell stepped up and calmly placed the resulting penalty past Brill.
Shortly after half-time the contest was effectively over when 18-year-old full-back Neal Eardley let fly from 20 yards and the ball took a vicious deflection off Chorley and past the helpless Brill into the goal.
Gillingham continued to push forward and looked to Matt Jarvis for inspiration but inevitably left gaps at the back that Oldham looked to expose.
Porter missed a chance to double his tally when he scuffed his shot from Eardley's cross, while Liddell also came close to adding a fourth.
But in the closing stages Gillingham were unfortunate not to grab a consolation through Jarvis, who saw his effort saved by Les Pogliacomi, while substitutes Brent Sancho and Danny Spiller narrowly failed to score at the death.