The Australian was two shots behind with two holes to play at Waialae, but made an eight-foot birdie putt on the final hole to force the play-off.
Smith, who shared the team title with Jonas Blixt in the Zurich Classic three years ago, sealed the victory with a two-putt par from 10 feet in the first play-off round.
Steele, a three-time PGA Tour winner, had taken a three-shot lead into the final round.
But he missed a six-foot par putt on the 17th and then hit a wild hook from the fairway on the par-five 18th.
Northern Ireland’s Graeme McDowell finished joint fourth on nine under. PA