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Ireland's Shane Lowry, ranked 68th in the world and the 64th-seed in the 64-man field, defeated Northern Ireland's McIlroy 1-up, while Charles Howell defeated Woods 2 and 1 at the Ritz Carlton Golf Club course at Dove Mountain.
Lowry, McIlroy's friend from their days in the amateur ranks, made his run on the back nine, chipping in for birdie to win the 12th hole and winning the 13th with an eagle to gain a two-up advantage.
"I was just trying to get up and down for the half, and it went in," Lowry said of his chip-in at 12. "That kind of got me going.
"Then I hit a lovely five-wood to five feet at the next and that got me going for the rest of the round."
McIlroy pulled one back at 14, but conceded the 15th after finding a series of bunkers. He then birdied the par-three 16th for a win, but they halved the last two holes with pars, and Lowry emerged victorious.
Lowry was in a fairway bunker off the tee at 18, and a greenside bunker with his second shot. McIlroy couldn't take advantage, however, hitting his second shot into the same greenside bunker.
McIlroy's bunker shot rolled within inches of the cup and was conceded par. Lowry blasted out to four feet and rolled in the match winner.
"I'm feeling quite good now," said Lowry, who gave a sigh after his match winner dropped. "But it's important not to get too high now, because it's only the first round, and I've got another match tomorrow and am really looking forward to playing that."
Neither Howell nor Woods made a bogey in their match, which ended with darkness closing in.
"I knew I had to play my best to have a chance out there," Howell said, adding that his mindset throughout was that he had "nothing to lose."
Howell never trailed after going 1-up with a birdie at the second. Woods squared the match with a birdie at the fifth, but Howell edged ahead at the eighth. Woods won the 13th with a birdie, but Howell regained a 1-up lead with a birdie at the 15th and held on.
He stretched his lead with a birdie to win 16. Woods couldn't birdie 17, and Howell's two-putt par gave him the win.
"He's the best of all time, I knew I had to play my best to even have a chance," Howell said. "Fortunately, I holed a few nice ones there at the end."
Woods, winner of this event in 2003, 2004 and 2008, has now failed to get out of the first round three times in his career, but said he couldn't complain about his game.
"We both played well," Woods said. "He made a couple more birdies than I did."
The round was continued from Wednesday, when a freak snowstorm in the Arizona desert cut short play. Thursday's start was delayed for several hours.
Two matches were halted by darkness. Sweden's Carl Pettersson was 1-up against Rickie Fowler through 17 holes with the winner to face Lowry.
Spain's Gonzalo Fernandez-Castano and Italian Francesco Molinari were all-square through 15, with the winner to face Howell.
McIlroy, who missed the cut in his only other start of the year - the European Tour's Abu Dhabi HSBC Golf Championship - is under extra scrutiny in the wake of his switch to Nike equipment.
He said some ragged iron play cost him the match.
"I drove the ball really well," McIlroy said, but "couldn't take advantage of that."
Still lurking in Lowry's quarter are major winners Jim Furyk, Graeme McDowell and Bubba Watson, who all advanced.
Howell's quarter includes US Open champion Webb Simpson and defending Match Play champ Hunter Mahan.
World number three Luke Donald, who won the title in 2011 to launch a stellar year that saw him gain the world number one spot and win the European and US money titles, advanced with a 1-up victory over Germany's Marcel Siem.
South African Louis Oosthuizen, the fourth overall seed and top of his quarter, defeated Scotland's Richie Ramsay 2 and 1.
First round scores on Thursday in the World Golf Championships Match Play Championship:
Shane Lowry (IRL) bt Rory McIlroy (NIR) 1-up
Carl Petterson (SWE) leads Rickie Fowler (USA) 1-up through 17, darkness halted play
Alexander Noren (SWE) bt Dustin Johnson (USA) 6 and 4
Graeme McDowell (NIR) bt Padraig Harrington (IRL) 2-up
Bubba Watson (USA) bt Chris Wood (ENG) 2 and 1
Jim Furyk (USA) bt Ryan Moore (USA) 4 and 2
Russell Henley (USA) bt Charl Schwartzel (RSA) 1-up
Jason Day (AUS) bt Zach Johnson (USA) 6 and 5
Hogan Bracket
Louis Oosthuizen (RSA) bt Richie Ramsay (SCO) 2 and 1
Robert Garrigus (USA) bt Branden Grace (RSA) 4 and 3
Marcus Fraser (AUS) bt Keegan Bradley (USA) 1-up
Fredrik Jacobson (SWE) bt Ernie Els (RSA) 1-up
Justin Rose (ENG) bt KJ Choi (KOR) 2 and 1
Nicolas Colsaerts (BEL) bt Bill Haas (USA) 5 and 4
Sergio Garcia (ESP) bt Thongchai Jaidee (THA) 20 holes
Matt Kuchar (USA) bt Hiroyuki Fujita (JPN) 3 and 2
Player Bracket
Charles Howell (USA) bt Tiger Woods (USA) 2 and 1
Gonzalo Fernandez-Castano (ESP) v Francesco Molinari (ITA) all-square through 15, darkness halted play
Webb Simpson (USA) bt David Lynn (ENG) 5 and 4
Peter Hanson (SWE) bt Thomas Bjorn (DEN) 3 and 2
Rafael Cabrera-Bello (ESP) bt Lee Westwood (ENG) 19 holes
Martin Kaymer (GER) bt George Coetzee (RSA) 2 and 1
Richard Sterne (RSA) bt Jason Dufner (USA) 1-up
Hunter Mahan (USA) bt Matteo Manassero (ITA) 5 and 4
Snead Bracket
Luke Donald (ENG) bt Marcel Siem (GER) 1-up
Scott Piercy (USA) bt Paul Lawrie (SCO) 4 and 3
Steve Stricker (USA) bt Henrik Stenson (SWE) 5 and 4
Nick Watney (USA) bt David Toms (USA) 5 and 4
Tim Clark (RSA) bt Adam Scott (AUS) 2 and 1
Thorbjorn Olesen (DEN) bt Jamie Donaldson (WAL) 3 and 2
Ian Poulter (ENG) bt Stephen Gallacher (SCO) 2 and 1
Bo Van Pelt (USA) bt John Senden (AUS) 6 and 5
