After an unspectacular first eight holes, Davis birdied the 9th and 10th to get to 14-under, then drew even with Molder at the 17th when he sank a 10-foot birdie putt, grabbing a piece of the lead at 16-under 194.
It was the first time the British European Tour veteran has been in first place after three rounds of a PGA Tour event.
Molder, the second-round leader, posted a 67 to keep a presence in the top spot. He and Davis finished one stroke ahead of former Masters champion Zach Johnson, who fired a 64 in round three.
Ben Crane (64), Jeff Overton (66) and Jason Bohn (68) shared fourth place at 14-under 196.
Swim Week Miami Beach 2014 [PHOTOS]