Goalkeeper Joe Willis had two saves for Nashville while posting his fifth shutout of the campaign.
It marked the second time this season that Nashville registered a 1-0 win over the Sounders. The other occasion was the season-opening match in Seattle on Feb. 27.
Nashville (8-6-6, 30 points), which improved to 3-1-4 at home this season, bounced back from Saturday's horrific 4-1 road loss to Charlotte FC.
Seattle (8-9-2, 26 points) played without All-Star Raul Ruidiaz (hamstring) for the fifth time in the past six matches.
Nashville had a 12-9 edge in shots and put five on target to Seattle's two.
The sequence for Nashville's goal began when Randall Leal stole the ball deep in the Seattle end and booted a long pass upfield.
C.J. Sapong won the race for the ball and then sent a centering pass to Mukhtar, and the All-Star sent a right-footed shot into the left corner for his 11th goal of the season.
Seattle had three solid opportunities to tie the match in a 17-minute span of the second half.
In the 58th minute, Danny Leyva was open, but his hard right-footed shot sailed wide left of the net.
Ten minutes later, Nouhou Tolo fired a left-footed shot that hit a defender, and Willis stuck up his right hand to slightly deflect it. The ball caromed off the post and back into the field of play.
Tolo delivered a stellar pass to a breaking Will Bruin in the 75th minute, but the latter's right-footed shot was just left of the net.
Nashville had a chance for a goal in the 26th minute when Sean Davis booted a left-footed shot to the left of Cleveland. But Cristian Roldan was standing at the goal line and prevented the ball from going in, and Sapong's rebound attempt was also stymied.