Allen's birdie on the last regulation hole gave him a round of 5-under 65, tying him with Browne at 13-under-par 197.
He then went on to record another birdie at Rock Barn, earning his fifth career win on the Champions Tour.
Browne had claimed a share of first place with final-round 6-under 64.
Claiming sole possession of third place at 11-under 199 was overnight leader Bernhard Langer, who stumbled to a 1-under 69 Sunday.
Swim Week Miami Beach 2014 [PHOTOS]