Josh Smith had 19 points and 13 rebounds, and Jeff Teague added 15 points in the opener of Atlanta's five-game road trip.
Brandon Jennings had 21 points and 11 assists to lead Milwaukee, and Mike Dunleavy added 17 points. However, the Bucks had no answer for Johnson in the fourth.
He tied the score at 84 with a 19-foot jumper. After the Bucks failed to connect on their next two possessions, Johnson put Atlanta ahead with a tiebreaking five-footer.
Drew Gooden hit a 3-pointer from deep in the corner for Milwaukee to tie the score at 88, but Smith hit a 3 and Johnson made two free throws to push the Hawks ahead for good at 93-90 with 16.6 seconds left.
Johnson, who came in averaging 18.6 points, was 10 of 22 from the floor and 8 for 8 from the line. He had five rebounds and two steals.
Zaza Pachulia grabbed a season-high 14 rebounds for Atlanta.
Andrew Bogut had six points and 12 rebounds and Gooden added 13 points for Milwaukee, which hoped to build on the momentum of big victories on the road.
The Bucks, who started 0-8 away from home, won 100-86 at New York on Friday and then 91-82 at Miami on Sunday despite shooting 35.0 percent.
With the Bucks on the attack, Marvin Williams deflected a pass from Dunleavy and the Hawks came up with loose ball. Milwaukee fouled Smith with 9.7 seconds left. He made two free throws and after Jennings raced in for a driving layup for Milwaukee's final points, Johnson made two free throws for the final margin.
Trailing 70-64 after three quarters, the Bucks opened the fourth with a 10-3 surge.
Dunleavy started it with an eight-foot jumper and then capped it with two free throws that gave Milwaukee the lead at 74-73 with 8:21 left. It was the 18th lead change in the game.
The Bucks recovered from shooting 9 of 23 (39 percent) in the first quarter with an 11-of-22 effort (55 percent) in the second to tie the score at 48 as the half ended.
Teague scored 11 points on 5-of-7 shooting for Atlanta, which shot 54.5 percent in the first quarter.
NOTES: Hawks G Kirk Hinrich has missed 17 games (left shoulder surgery), but made the trip. He has been practicing hard, according to coach Larry Drew, and may return to action during the road trip. ... Oak Hill Academy (VA) was well represented at the game. Atlanta's Josh Smith and Jerry Stackhouse and Milwaukee's Brandon Jennings and Stephen Jackson all graduated from there. ... Milwaukee Brewers general manager Doug Melvin and his wife, Ellen, were in attendance.