The Hawks, coming off consecutive road losses at Orlando and Oklahoma City, improved to 20-5 at home.
Johnson began the second half going 7 for 8 from the field and finished 11 for 21, with three 3-pointers.
"We picked up our energy," Johnson said. "I'm just glad we came through."
Chris Kaman and Eric Gordon each scored 17 points for Los Angeles. The Clippers finished an eight-game trip with just two wins and have dropped 13 of 15 away games.
Marcus Camby, the NBA's third-leading rebounder, had 20 rebounds for Los Angeles, but a miscommunication with Baron Davis caused the point guard to call the Clippers' final timeout with 2:16 remaining.
Camby apparently didn't break free soon enough to help Davis move the ball away from two defenders that had him pinned near the midcourt sideline.
Because he feared committing a 5-second turnover, Davis called timeout. He complained to Camby, who nodded his head in agreement, as the two teammates walked to Los Angeles' bench.
The possession ended as Davis badly missed a layup from the right side.
"I think they were the more aggressive team in the second half," Davis said. "They started making shots, which helped them get confidence. A good team like this is hard to contain when they start knocking down shots."
Camby was disappointed to end the road trip with another loss.
"It's definitely frustrating," Camby said. "We felt we let this one get away. They cut the lead to 3 going into the fourth. Going down the stretch they got to the line a lot and made a lot of free-throws."
Atlanta never trailed after Crawford's two free throws made it 90-88 with 3:35 left.
The Clippers were up 70-57 after Rasual Butler's fast-break layup at the 2:16 mark of the third quarter. Their defense, however, relaxed enough to let Atlanta go on a 13-0 run that ended with Crawford's free throw.
"We just got up and them put them on a carousel," Hawks forward Josh Smith said. "We limited them to one shot, rebounded and had a couple of fast-break points. That is what this team has to do."
Smith, Horford and Marvin Williams each had 10 rebounds. Horford finished with 16 points, and Smith had 15.
Though the Hawks were outrebounded by nine and outscored by 16 in the paint, Horford believes his team fought harder for loose balls.
"It's very important," Horford said. "Josh is so aggressive out there, and we all need to do that as a unit in order to be effective on the boards."
Horford and Johnson combined for 24 points and hit 10 of 11 free-throw attempts in the fourth quarter.
"We can't afford to have a letdown," Johnson said.
The Clippers dropped to 2-15 when starting the fourth quarter with a deficit. ... Atlanta improved to 18-6 when Crawford, who went 8 for 16 from the field, hits at least 50 percent of his shots. ... Though Los Angeles won last year at Philips Arena, it dropped to 13-39 in Atlanta. ... The Hawks lead the series 67-36.