The 2010 title winner was one of three leading English players to fall at the first hurdle. Poulter lost 2 and 1 to Rickie Fowler, Lee Westwood was beaten 5 and 3 by Harris English, and Luke Donald went down 5 and 4 to Matteo Manassero.
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Rory McIlroy defeated Boo Weekley 3 and 2 while his fellow Northern Irishman Graeme McDowell beat Gary Woodland one-up after 19 holes.
After the opening three holes were halved in his match with Poulter, Fowler won three of the next four and needed only one birdie to do so, although Poulter then reduced his deficit with a birdie on the eighth.
A par on the 10th was enough for Fowler to move three-up again and although he gave the next hole back with a bogey six, a superb eagle on the par-five 13th - with Poulter in trouble over the green - restored a three-hole cushion.
Poulter typically refused to throw in the towel and won the 14th with a birdie from 18ft as Fowler missed from much closer, but further birdie chances went abegging on the next three holes and his fate was sealed. Poulter posted on Twitter: "Disappointed with my performance today, did not deserve to move on. Congrats to Rickie Fowler."
Fowler said: "I knew it was going to be a tough match. Ian wasn't playing great today but I knew he was going to find a way to hang around and give me something at the end. Obviously it feels good to get a win against him and I am excited for tomorrow."
Finland's Mikko Ilonen had looked on course to cause the first upset when he led Bubba Watson by two holes with six to play, but he lost the next three and Watson, who claimed his first win since the 2012 Masters in the Northern Trust Open on Sunday, sealed a 2 and 1 win.
"After [hole] number 11 I knew I had to fight hard and somehow I did it," Watson said. "It's very tough with all the media and attention around me after the win on Sunday to focus enough on golf so hopefully this match scared me enough that I will focus a little harder."
Matt Kuchar, the defending champion, also advanced to the second round, but only after having to win his match with Austria's Bernd Wiesberger twice. Kuchar thought he had sealed a 5 and 4 win on the 14th but had failed to move his ball marker back to its correct position before holing his birdie putt. That cost him the hole and meant the match continued, but the world No.10 still had a three-hole cushion and eventually won 3 and 2.
There was better news elsewhere for two more of the 25-strong European contingent, with Peter Hanson beating Dustin Johnson 4 and 3, and Sweden's Jonas Blixt moving into the last 32 with a 2 and 1 win over the former US PGA champion Keegan Bradley. The South African George Coetzee beat Steve Stricker 3 and 1. Stricker's participation had been in serious doubt until his brother underwent a successful liver transplant at the weekend.
Billy Horschel of the USA thrashed the Welshman Jamie Donaldson 6 and 5, Bill Haas beat Miguel Angel Jimenez 5 and 4, and Thomas Bjorn defeated Francesco Molinari 2 and 1. Sergio Garcia required 22 holes to see off the Australian Marc Leishman, the Spaniard making a winning birdie on the fourth extra hole.