Mukhtar scored in the 51st minute to secure a spot in the final four against the winner of the series between the Seattle Sounders and the UANL Tigres.
The first leg ended with a scoreless draw a week ago in Nashville.
Club America, a seven-time Champions Cup winner, was aiming for its first regional title since 2016.
In the second half, referee Walter Lopez stopped the game a couple of times when Club America fans shouted a homophobic chant against goalkeeper Brian Schwake.
Dennis Bouanga scored a late penalty to help LAFC draw 1-1 with defending champion Cruz Azul and advance to the semifinals.
Gabriel Fernandez converted a penalty kick in the 18th to put the home team ahead, but Bouanga equalized with a right-foot shot from the spot deep in stoppage time.
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LAFC, which won the first leg 3-0 with two goals from David Martinez, will play the winner of the series between Toluca FC and LA Galaxy.
“I know we eliminated a great club, a great champion, and a team that gave us a lot of trouble. Qualifying was tough,” LAFC coach Marc Dos Santos said. “Now there are no favorites. Everyone who makes it to the semifinals is strong.”
Former Tottenham goalkeeper Hugo Lloris finished the match with eight saves for LAFC, which is aiming for its third final in the tournament after losing in the championship deciders in 2020 and 2023.
The eight saves from the 39-year-old Frenchman were the most in his 102 matches with the LAFC in all competitions.
Cruz Azul finished the match with 10 men after Gonzalo Piovi was sent off in the 90th minute after tackling Bouanga.
“Today we had 31 shots, we faced a world champion goalkeeper who prevented us from having a great night,” Cruz Azul coach Nicolas Larcamon said.
The Mexican club, a seven-time Champions Cup winner, was aiming to emulate Club América, the last team to clinch back-to-back titles in the 2014-15 and 2015-16.
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