The Celtics have lost five of seven, including losses to Orlando and Atlanta in back-to-back days.
"I don't really know if we could have won the game tonight, the way the Hawks were playing," said Boston coach Doc Rivers. "Especially coming off a back-to-back, this is a tough team to play."
Pierce said the Celtics "play in spurts."
"That seems to be the problem right now," Pierce said. "Overall, I liked our fight tonight, but it wasn't enough."
Kevin Garnett had 15 as Boston's only other scorer in double figures.
Johnson scored nine of the Hawks' first 11 points of the fourth period.
"I just wanted to assert myself and be aggressive and make plays," Johnson said.
Johnson said Crawford, Josh Smith (12 points, 9 rebounds, 3 blocks) and Al Horford (11 points, 8 rebounds) carried the team in the first three periods.
Crawford said Johnson "turned the switch on," in the final period.
"He's fun to watch when he gets it going like that," Crawford said.
Crawford made three 3-pointers -- including a halfcourt shot to end the first period -- and led the team with six assists.
"We can look anybody eye to eye," Crawford said. "We can stand up. I think we're going to get better. This shows our growth."
The Hawks were celebrating with their sellout crowd and most Boston players were already off the court as the final seconds ticked off the clock.
"That's the kind of atmosphere you want, a playoff atmosphere," Crawford said. "I was kind of fired up."
Johnson, named to his fourth straight All-Star team on Thursday, made only 3 of 13 shots from the field in the first three quarters before finding his touch to help the Hawks fend off the Celtics' comeback. Atlanta led 74-73 at the start of the fourth before pushing the advantage to 11 points.
The Hawks' last season sweep of the Celtics came with three wins in 1998-99. Their last 4-0 sweep was in the 1995-96 season.
The Hawks completed the sweep one day after having two players -- Johnson and Horford -- named to the All-Star team for the first time since 1998.
Emotions rose in the third period soon after Hawks coach Mike Woodson began complaining that his team had been called for five fouls while Boston had none.
With 7:42 remaining in the period, Smith was called for a flagrant foul and Boston's Kendrick Perkins drew a technical after Smith grabbed Perkins from behind to stop a shot. Players from both teams swarmed around the two.
Woodson drew a technical foul from official Tony Brothers after complaining about the flagrant foul on Smith. Zaza Pachulia drew another technical for protesting after he was called for a foul with 2:50 remaining in the period.
The Celtics made 19 of 20 free throws in the third to cut into Atlanta's lead.
"They didn't win a championship not being a physical team," Woodson said. "They're still a great team defensively and they get after you."
The Hawks led 53-41 at halftime despite a slow start from Johnson, who made only one of eight shots in the half.
Crawford's halfcourt shot to end the first period gave Atlanta a 23-22 lead -- its only lead of the period.
The reported attendance of 18,732 was the Hawks' fourth home sellout of the season. ... Rasheed Wallace was called for his NBA-leading 13th technical foul in the second period. ... G J.R. Giddens, now with Maine of the NBA Development League, saw a doctor on Friday and is facing surgery on his left knee. The Celtics assigned Giddens to Maine on Jan. 21. ... The Celtics swept the 2008-09 season series 4-0.