The win handed Gillingham their third successive victory and ended the Swans' five-game unbeaten league run.
After a tentative start by both sides, Gillingham began to take the initiative and fashioned the first chance on six minutes when Clint Easton drove wide from 12 yards after the ball had deflected off Matt Jarvis in the area.
They did not have to wait long for the opener however. Easton's short corner was swung into the area by Danny Jackman and fellow defender Cox was on hand to sweep the ball home at the far post.
Swansea were rattled and matters became worse in the 30th minute when Michael Flynn released Jarvis allowing the youngster to run at Izzy Iriekpen before firing a fine shot into the top left-hand corner.
Three minutes later Easton forced a superb save out of keeper Willy Gueret from a free-kick as the Gills threatened to run riot.
But on the stroke of half-time giant defender Dennis Lawrence should have scored when he rose to meet Andy Robinson's free-kick but directed his header wide from six yards.
After Jarvis missed a decent opportunity to extend the lead shortly after the restart, Lawrence almost made them pay when he headed Andy Robinson's corner wide.
Swansea manager Kenny Jackett clearly needed to act and introduced three substitutes in former Gillingham man Tom Williams, midfielder Darren Way and left winger Thomas Butler.
Butler attempted to inspire his side and in the 68th minute his cross found Adebayo Akinfenwa in space allowing the striker to turn and fire his shot against the outside of the post.
But the contest was effectively over in the 79th minute when Swansea failed to clear Easton's corner and the ball fell invitingly to Cox who smashed home his second of the game.
Swansea refused to give in and snatched a consolation goal through Leon Knight five minutes from time.
The young striker finished acrobatically after seeing his first effort brilliantly saved by Scott Flinders following Akinfenwa's flick.
Jarvis should have added some gloss to the scoreline but saw his shot rebound off the post.