Arizona remained unbeaten in its 21 games this season, holding Stanford to just one field goal during the game's final 10 minutes.
A jump shot by Josh Huestis with 10:09 remaining gave Stanford a 53-49 lead and the Cardinal did not score another point until Dwight Powell made a reverse layup with 1:26 to go.
Despite that long scoring drought, Powell's basket tied the game at 55-55. Johnson, however, hit a 3-pointer with 51 seconds left that put the Wildcats in front for good and his two foul shots with six seconds remaining accounted for the final score.
Chasson Randle then missed a 3-point try for Stanford with two seconds showing on the clock. Stanford went 1-of-13 from the field during the closing 10 minutes.
Johnson finished with 16 points and T.J. McConnell had 11. Arizona climbed to 8-0 in the Pacific-12 Conference while every other team in the league has at least two losses.
Powell and Huestis both had 13 points for the Cardinal (13-7, 4-4 Pac-12).