After last week's narrow miss at the Dell Accenture WGC Match Play event, Poulter went into this week's tournament knowing that only victory would do to claim the last spot at Augusta next week.
The Englishman led for the majority of the final day's play, but Hossler registered four successive birdies on the back nine to put himself at the top of the leaderboard heading into the closing holes.
The 23-year-old American came within millimetres of winning his first tournament on the PGA Tour with a 30-foot putt on the last, before then watching Poulter hole from just under 20 feet to force a playoff.
Both men played reasonable drives at the first extra hole but Hossler sent his second into the bunker before mishitting his third shot into the water, effectively leaving Poulter with three putts to win.
After Hossler ended with a seven, Poulter made par to seal a win which takes him up to 29th in the World Golf Rankings.