After a poor opening day 75, McDowell has rediscovered some good form with back-to-back rounds of six under par.
He fired five birdies in a row on Saturday to finish the day on eight under for the tournament, 10 shots behind leader Gary Woodland of America.
Waterford's Seamus Power is in fourth place on 15 under after a round of 64.
Defending champion McDowell just made the cut on two under on Friday after a much-improved second round in Playa del Carmen.
Starting on the 10th tee on Saturday, he birdied two of his opening six holes but dropped his only shot of the day on the par-four 16th.
A flurry of birdies on his back nine helped him move up the leaderboard to eight under and into a tie for 32nd place.
McDowell has had five top-10 finishes in 2016 but some poor results this year has seen him fall to 81 in the world rankings.
Graeme McDowell carded a 65 for the second day running in an encouraging defence of his PGA Tour title in Mexico.