Fleetwood's 5-under round was enough to catch Gonzalez, who has been atop the leader board all three days of the tournament despite a wobbly 70 on Saturday.
The leaders were tied at 16-under 200 after 54 holes. Saturday's round included a lengthy suspension due to lightning and rain.
Fleetwood turned in a pair of birdies and an eagle on the par-5 second hole at Gleneagles. He needed them all to offset three bogeys, including one on the final hole.
Gonzalez bogeyed the first hole but followed it up with three birdies on the front nine. He cooled off after the turn with one birdie and one bogey to finish 2-under for the day. The Argentine shot 65 in his first two rounds.
Swim Week Miami Beach 2014 [PHOTOS]