The Hawks squandered a 20-point lead in the fourth quarter, but made 10 free throws in the final minute to earn their fourth victory in five games.
Josh Smith had 14 points and 11 rebounds as Atlanta beat the Kings for the sixth straight time. Former Sacramento star Mike Bibby added nine points and 10 boards.
Tyreke Evans had 15 of his 29 points in the fourth quarter for the Kings, who outscored Atlanta 39-31 in the final period. DeMarcus Cousins also had a huge fourth, scoring 16 of his 24 points.
Sacramento, which owns the NBA's worst record at 7-25, closed to 106-102 with 8 seconds remaining, but Crawford made a pair of free throws to secure the victory.
The Kings rallied in the fourth quarter. Evans had a three-point play and Carl Landry made two free throws to cut the Atlanta lead to 96-89 with 1:40 left.
Back came the Hawks. After Evans turned a steal into a layup, Johnson converted two free throws with 44 seconds left to put Atlanta ahead 100-93. Evans missed a 3-pointer and Crawford got the rebound and was fouled. His two free throws made it a nine-point cushion.
Crawford hit three 3-pointers and scored 15 points in the opening half to help the Hawks to a 54-40 lead. Atlanta scored 11 unanswered points late in the second quarter.
NOTES: Atlanta coach Larry Drew was a guard on the first Sacramento team in 1985-86. ... Bibby, the NBA's third-leading 3-point percentage shooter at 49 percent, made three of Atlanta's seven 3-pointers in the first half. ... Bill Walton, an occasional member of the Kings' broadcast team, did color commentary for both radio and TV. ... The Hawks conclude a four-game trip Wednesday in Utah, then play four of five at home where they have won nine straight.