Poulter, who won the title in 2010, lost 2&1 to American Rickie Fowler, who led from the fourth hole onwards and will now face compatriot Jimmy Walker in the second round.
Walker, who has already won three times in the 2014 PGA Tour season, secured the first victory of the day with a comfortable 5&4 win over South African Branden Grace.
After the opening three holes were halved, 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.
In typical fashion Poulter 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 begging on the next three holes to seal his fate.
"Disappointed with my performance today, did not deserve to move on," Poulter wrote on Twitter. "Congrats to Rickie Fowler."
Fowler said on Sky Sports: "I knew it was going to be a tough match. He 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 like creating a shock when he led Bubba Watson by two holes with six to play, but lost the next three holes and Watson - who claimed his first win since the 2012 Masters in the Northern Trust Open on Sunday - sealed a 2&1 win on the 17th.
"After number 11 I knew I had to fight hard and somehow I did it," Watson said. "It's very tough will 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."
Defending champion Matt Kuchar 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&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 number 10 still had a three-hole cushion and eventually won 3&2.
There was better news elsewhere for two more of the 25-strong European contingent, with Peter Hanson beating Dustin Johnson 4&3 - the world number eight's fifth first-round loss in six years - to set up a meeting with France's Victor Dubuisson, who saw off Kevin Streelman 5&4.
Sweden's Jonas Blixt moved into the last 32 with a 2&1 win over former US PGA champion Keegan Bradley and South African George Coetzee saw off Steve Stricker 3&1. Stricker's participation in the event had been in serious doubt until his brother underwent a successful liver transplant at the weekend.
The biggest margin of victory so far belonged to American Billy Horschel, who thrashed Welshman Jamie Donaldson 6&5, while Bill Haas beat Miguel Angel Jimenez 5&4 and Thomas Bjorn defeated Francesco Molinari 2&1.
Sergio Garcia required 22 holes to get the better of Australian Marc Leishman, the Spaniard making a winning birdie on the fourth extra hole, while in the later matches Rory McIlroy was four up on Boo Weekley after 11.
A last-32 clash with Lee Westwood looked unlikely however, the former world number one four down to Harris English after 10.
Westwood was still four down with four to play, but could take heart from the unlikely victory produced by Graeme McDowell from a similar position.
McDowell was four down after seven holes against Gary Woodland and three down with three to play, but won the 16th with a par, the 17th with a birdie and was conceded the 18th as Woodland went from one greenside bunker to another.
A birdie on the first extra hole then sealed victory for McDowell, who reportedly shared a private jet from Florida to Arizona with his opponent.
There was no such escape for Luke Donald however, the 2011 winner and former world number one being thrashed 5&4 by Italy's Matteo Manassero.