The world No 1 was one of only four players to finish under par in strong winds at Shinnecock Hills, posting a one-under 69 to join Ian Poulter, 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 only fared slightly better as he triple-bogeyed his opening hole on his way to an opening-round 78.
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.
Henley was briefly handed the outright advantage when Johnson bogeyed the 12th and 14th, only to miss from 10-footer 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.