Stroud, who had four consecutive birdies in a front-nine stretch, shot a 4-under-par 68 while Moore, who suffered through a back-nine string in which he had four bogeys in five holes, had a 69. They are both at 12-under 204 after 54 holes.
Kiradech Aphibarnrat, whose round included an eagle and a double bogey, turned in a third-round 69 and Gary had birdies at 15 and 16 in his 67. That left them tied for third at 11-under. Jerry Kelly (66 Saturday) is alone in fifth at 10-under.
Keegan Bradley, who was up by four strokes after 36 holes, sagged to a 4-over 76 in the third round and slid into a tie for sixth with Graham DeLaet (68) and Stewart Cink (69).
Moore was 6-under through 10 holes Saturday but had bogeys at Nos. 12 and 13 and, after a birdie at 14, two more bogeys at 15 and 16.
That last bogey resulted in a two-stroke swing with Stroud, who had a birdie on the hole.
Stroud opened his day with a birdie and made his move up the leaderboard with four straight birdies over Nos. 5-8. He gave back two of those strokes at 12 and 15 before the birdie at 16 tied him with Moore for first.
Swim Week Miami Beach 2014 [PHOTOS]