The Tigers bounced back from a home loss to Cincinnati last weekend to produce a key road victory over the defending national champions.
The lead changed hands eight times in the second half and Louisville took its last advantage with 2:09 to play at 67-66 on a layup by Montrezl Harrell.
Louisville did not score another point, missing its last four shots and suffering a key turnover as well.
Johnson's two free throws with 1:24 remaining pushed the Tigers in front for good, Johnson added a layup and Joe Jackson hit a foul shot with 18 seconds remaining to give Memphis a 71-67 lead.
Shaq Goodwin hit a breakaway dunk shot just before the buzzer to complete the scoring and lift Memphis to 10-4 for the season. The Tigers are 2-1 in the American Athletic Conference.
Louisville dropped to 13-3 for the year and to 2-1 in the league, leaving Cincinnati (3-0) as the only team without a conference loss.
Jackson and Goodwin both had 15 points and Johnson added 13 points and 11 rebounds for the Tigers, who shot 50 percent from the field. Louisville was led by Russ Smith with 19 points, but he made only 6-of-15 field goal attempts.