Johnson played his last seven holes knowing he had to review a possible rules infringement after the round.
He shot 68 to finish five under but was then penalised a shot for making his ball move on the fifth green, despite being initially absolved of wrongdoing.
McIlroy called organisers "amateur" and Spieth said the decision was "a joke".
Johnson's score was amended to four under, but he still finished three shots ahead of Irishman Shane Lowry and American duo Jim Furyk and Scott Piercy at Oakmont.
