Trailing 1-0 after losing the first leg at the Stubhub Center, Tijuana scored twice in the opening nine minutes through Jaime Ayovi at the Estadio Caliente.
Robbie Keane twice put the Galaxy to within a goal of the last four but they were unable to find another despite their best efforts late in the encounter.
The win continued Liga MX side Tijuana's stunning run at a tournament they are appearing in for the first time.
Cesar Farias' men made a dream start in front of their home fans and levelled the tie after just 57 seconds.
It was a scrappy opener from Ayovi, who tucked away a finish at the back post after two team-mates had failed to put away their earlier chances.
Less than 10 minutes in and Tijuana were ahead in the tie as Ayovi got on the end of a Javier Guemez pass, went past goalkeeper Jaime Penedo before passing into an open goal.
The tie looked over after 26 minutes, when Dario Benedetto beat his marker to a loose ball before drilling a finish into the bottom corner from just inside the area.
But Keane kept the Galaxy alive when he tapped in Omar Gonzalez's squared header just after half-time, becoming the club's all-time leading goalscorer in the CONCACAF Champions League with six.
Bruce Arena's side pushed for another but were caught out as Richard Ruiz finished tidily from an angle after Benedetto flicked on a header in the 82nd minute.
Once again, Keane dragged the Galaxy back into the tie as he placed a Landon Donovan pass into the bottom corner three minutes later.
Los Angeles, who reached the semi-finals last season, continued to push but were unable to find another goal.
