World number one Johnson joined fellow Americans Russell Henley and Scott Piercy and Englishman Ian Poulter at the top, with Jason Dufner a stroke behind on a day when strong winds helped send the scoring average soaring to 76.4.
Among those who succumbed to the conditions were Woods (78), Spieth (78) and McIlroy (80) while Phil Mickelson, who needs victory to complete the career grand slam, missed one fairway all day but could not conjure up his usual deft touch around the greens. He finished with a 77.
Spieth and Woods each made an early triple-bogey, while McIlroy was 10-over after 11 holes en route to his highest U.S. Open score.
Johnson's scariest moment came after a poor drive at the sixth disappeared into the long fescue rough. The ball was only found when former PGA champion Rich Beem, an on-course commentator for British television, trod on it.
"You had to be patient and stay focused all day, which I felt like I did a really good job (of) today," Johnson, the 2016 champion, said.
"I felt like I did everything pretty well today. I drove it well, hit a lot of quality iron shots, and then my short game was good.
"I knew before we even started the round that today was going to be very tough.
"It was hard, but it was fair. It wasn't anything tricky or unreasonable by any means."
Joint leader Poulter put a wretched U.S. Open record behind him for his best first-round score. He made his U.S. Open debut at Shinnecock Hills 14 years ago and it has been a miserable ride ever since.
"I didn't enjoy it at all, I have to say," said the European Ryder Cup stalwart.
"I haven't enjoyed very many, to be honest. They're difficult, they're hot, they're stressful.
"Feels like you're pulling teeth every single hole you play."
However, the 42-year-old realises that the U.S. Open is tough for everyone and has come into the tournament with a different attitude this year.
"I've changed my mindset. I'm here to enjoy my golf this week, to play freely.
"If I hit it in the rough, I hit it in the rough. I'm going to try and make par the hard way. It's difficult for everyone."
Any hopes that Shinnecock Hills would yield some good early scores were dashed as a strong wind whipped across the treeless layout from the moment the first groups teed off just before 7 A.M. local time (1100 GMT).
The carnage was widespread but nobody suffered more than English sectional qualifier Scott Gregory, who managed only three pars in a 92, the day's worst score by five strokes.
Winds are forecast to abate somewhat over the final three rounds but still blow strongly enough to ensure a stern test.
(Editing by Peter Rutherford)
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