Austrian Bernd Wiesberger upset Johnson, beating the big-hitting American 3&1 at the start of the group phase at Austin Country Club.
That shock came after McIlroy had also tasted defeat, losing by 2&1 to former European Tour Rookie of the Year Peter Uihlein.
“At the end of day, we both didn’t have a great day,” said Wiesberger after his victory. “He gave me a couple of holes, I gave him a few. At the end of the day, I’m glad that I prevailed and got a win.
“It wasn’t his best day, it wasn’t the best of my days. I think we both have to do a little bit of work this afternoon.”
McIlroy came into the event, which he won in 2015, on the back of a thrilling victory in the Arnold Palmer Invitational after a closing round of 64.
But the world No 7 found himself five down after just 10 holes and although he birdied five holes in succession on the back nine to stay alive, Uihlein breathed a sigh of relief after a par on the 17th to seal victory.
McIlroy could still advance to the last 16 with wins over Jhonattan Vegas today and Brian Harman tomorrow, although that would not be enough if Uihlein goes on to win all three group matches.
Johnson, who never trailed in any of his seven matches last year, was one up after five, but ran up a quadruple-bogey nine on the sixth and a double bogey on the 11th on his way to a surprise defeat.
World No 2 Justin Thomas was earlier made to work hard for victory against Luke List, despite his opponent giving himself a self-inflicted handicap.
List was forced to putt with his sand wedge after bending his putter out of shape in frustration during the early stages of the Group Two contest.
But the world No 67, who lost out to Thomas in a play-off for the Honda Classic last month, battled back from three down with four to play to take the tie all the way to the 18th before Thomas finally completed just his second victory in seven matches in the event.
Thomas, who can become world No 1 with a victory this week, admitted the thought of an embarrassing defeat had crossed his mind, telling PGA Tour.com: “I was thinking about it, how bad that would hurt if I couldn’t get it done. I don’t know what happened (with List’s putter) but I knew that it would hopefully make it easier on me. But it didn’t.”
List explained what had happened, adding: “I was walking off the sixth green and I was a little unhappy about the way I was feeling, a little under the weather.
“I thought it was a brush area and I just kind of swiped my putter and it turned out to be a wall. Stupid on my part.”
The other match in Group 2 saw Italy’s Francesco Molinari beat Patton Kizzire 3&1, while England’s Tyrrell Hatton beat France’s Alexander Levy 3&2 in Group 12.
Reverting to the putter he used to spark the ‘Miracle at Medinah’ in the 2012 Ryder Cup paid dividends for Ian Poulter, who beat European No 1 Tommy Fleetwood 3&2 in an all-English clash in Group 9.
“My stats over the last number of months have been better than I’ve seen before, tee to green, so if I can start to hole the putts I’ve holed with that putter I’m going to do some good things,” Poulter told Sky Sports.
“I’m in need of a really good week this week to get into Augusta [for the Masters] so it’s about getting the feeling back with something I’ve had success with. I can’t blame the putter, I hate doing that. If I grab hold of a putter that I know has worked, then it’s on me.”
Phil Mickelson was another surprise loser on day one as Charles Howell won their match 3&2, but Valspar Championship winner Paul Casey recovered from two down with seven to play to beat Russell Henley on the 18th.
Casey, who never led until winning the final hole, told Sky Sports: “I feel a little guilty. I should be wearing a mask but I’m very happy with it and can build on it during the week.”
Masters champion Sergio Garcia, whose wife gave birth to their first child last week, birdied the 18th to beat India’s Shubhankar Sharma by one hole.
(US unless stated)
Pat Perez halved with Si Woo Kim (Kor)
Gary Woodland halved with Webb Simpson
Justin Thomas bt Luke List 2 up
Francesco Molinari (Ita) bt Patton Kizzire 3 & 1
Tyrrell Hatton (Eng) bt Alexander Levy (Fra) 3 & 2
Charley Hoffman Lost to Brendan Steele 1 down
Hideki Matsuyama (Jpn) bt Yusaku Miyazato (Jpn) 2 & 1
Patrick Cantlay Lost to Cameron Smith (Aus) 2 down
Alex Noren (Swe) bt Kevin Na 4 & 2
Tony Finau bt Thomas Pieters (Bel) 2 & 1
Jordan Spieth bt Charl Schwartzel (RSA) 2 & 1
Patrick Reed bt Haotong Li (Can) 3 & 2
Ian Poulter (Eng) bt Tommy Fleetwood (Eng) 3&2
Kevin Chappell bt Daniel Berger 3&2
Jason Day bt James Hahn 4&2
Louis Oosthuizen (RSA) bt Jason Dufner one hole
Matt Kuchar halved with Zach Johnson
Yuta Ikeda bt Ross Fisher 2&1
Bernd Wiesberger (Aut) bt Dustin Johnson 3&1
Kevin Kisner halved with Adam Hadwin
Julian Suri bt Marc Leishman (Aus) 3&2
Bubba Watson bt Branden Grace (RSA) 5&3
Peter Uihlein bt Rory McIlroy (NIre) 2&1
Brian Harman halved with Jhonattan Vegas (Col)
Charles Howell bt Phil Mickelson 3&2
Rafael Cabrera-Bello (Spa) bt Satoshi Kodaira (Jpn) 2 & 1
Jon Rahm (Spa) halved to Keegan Bradley
Kiradech Aphibarnrat (Tha) bt Chez Reavie 3 & 2
Paul Casey (Eng) bt Russell Henley 1 up
Matthew Fitzpatrick (Eng) Lost to Kyle Stanley 1 down
Sergio Garcia (Spa) bt Shubhankar Sharma (Ind) 1 up
Xander Schauffele bt Dylan Frittelli (Rsa) 1 up