Mukhtar's second and third goals of the season helped Nashville (2-0-5, 11 points) continue its unbeaten start to the season, and marked the third time his side has overcome a two-goal deficit to earn a point.
The previous two came at home in the first two weeks of the season, in 2-2 draws against Cincinnati and Montreal.
Marcelino Moreno scored his third goal of the season and Erik Lopez added the first of his MLS career for Atlanta (2-1-4, 10 points), which was held without a victory for the first time at home this season.
The hosts were largely in control for most of the match, with goalkeeper Brad Guzan having to make only two saves to keep Nashville scoreless prior to Mukhtar's stunning rally.
The first of Mukhtar's goals came in the 80th minute only a few seconds after Guzan had extended his right leg to deny CJ Sapong's effort.
Nashville kept possession after the save, and fullback Daniel Lovitz played a give-and-go with midfielder Matt LaGrassa down the left. Then he crossed into the box for Mukhtar, who hammered a low finish beyond Guzan and inside the right post.
It was LaGrassa involved again in the equalizer, taking a pass from Sapong on the right and striking a shot that deflected off Atlanta's Miles Robinson.
Mukhtar was first to the rebound and hammered a low finish, this time between Guzan and the left post, to make it 2-2.
Moreno opened the scoring after only six minutes when he settled a long diagonal ball from Robinson, outmuscled two defenders to keep possession and then struck a low finish that beat Joe Willis.
It was 2-0 after six minutes of the second half. Moreno was involved again, this time playing with George Bello down the left. Bello crossed toward the near corner of the 6-yard-box, where Lopez redirected it past Willis with a first-touch finish off the back of his heel.