Poulter and world No 1 Johnson were two of only four players to finish under par in strong winds at Shinnecock Hills, posting one-under 69s to join Russell Henley and Scott Piercy at the top of the leaderboard.
Jason Dufner leads the chasing back after holing a 12-footer at the last to post a level-par 70, while Justin Rose and Henrik Stenson sit in the group of seven players two strokes off the pace.
McIlroy equalled the worst round of his major career as he slumped to a 10-over-par 80, while Woods fared only slightly better as he triple-bogeyed his opening hole on his way to an opening-round 78.
Poulter made a bright start when he rolled in a huge 45-foot putt for birdie at the third, and he holed another from half that range at the seventh before a poor second to the 10th resulted in his first bogey.
But he responded in style and came agonisingly close to a hole-in-one when his ball hit the pin at the short 11th, although he got into trouble at the 13th and did well to escape with just a bogey before he parred safely in to share the early clubhouse lead.
Johnson holed from 10 feet to save par on each of his opening two holes before following a 12-foot birdie at the third by nailing a putt from double the distance at the par-four next.
The 2016 champion cancelled out a bogey at the sixth by holing out from a bunker at the eighth to reach the turn in 33, before holing a 20-footer at the 12th to move ahead.
“I’m very pleased with the round,” said the in-form Johnson at the Sky Cart afterwards. “Anything under par on this golf course is very good, especially in the conditions we had today. From start to finish, it was very difficult.
“You had to focus on every single shot you hit, putts, everything. You know, it was just difficult all day, and you really had to be patient and focused, which I felt like I did a really good job of. But I felt like the USGA did a great job with the set-up. I felt like it was hard, but it was fair. It wasn’t anything tricky or unreasonable by any means.”
Henley was briefly handed the outright advantage when Johnson bogeyed the 12th and 14th, only to miss from 10 feet to save par at the last and slip back into the group on one under.
McIlroy dropped six shots in a four-hole stretch and bogeyed two of his final three holes to all-but end his hopes of a fifth major title, while playing partners Jordan Spieth and Phil Mickelson posted rounds of 78 and 77 respectively.
Jon Rahm shot an opening-round 78 and Jason Day posted a nine-over 79, while Scott Gregory propped up the leaderboard after racking up a 92
Watch the US Open throughout the week live on Sky Sports. Live coverage continues on Friday with Featured Group action from 1pm on Sky Sports US Open.