Galaxy newcomer Samuel Rosa Goncalves, a Brazilian forward known simply as Samuel, scored in the 11th minute in front of 15,159 at StubHub Center.
The Galaxy defense, led by Panamanian goalkeeper Jaime Penedo, spent the rest of the game holding back a Club Tijuana attack that became increasingly zealous in the second half.
The second game between the Galaxy and the Xolos is Tuesday in Tijuana and the winner will be decided by the number of aggregate goals.
"Winning at home is important," Galaxy Coach Bruce Arena said. "Getting a clean sheet is important. We're not in a great position but we're in a good situation.
"Obviously another goal would have benefited us but it's nice going in there with an advantage."
Samuel, 23, who joined the Galaxy on loan from Brazilian club Fluminense, scored after a pass bounced hard off teammate Robbie Keane's chest and toward Samuel inside the 18-yard box.
Samuel was running toward the goal when the ball went his way, and he then drilled it just above the ground about 12 yards past goalkeeper Cirilo Saucedo.
The Xolos' offense was fairly quiet until it went on the attack in the second half. All five of Club Tijuana's shots on goal occurred in the latter 45 minutes.
In the 52nd minute, Club Tijuana defender Hernan Pellerano struck a header from close range and Penedo made a superb diving save, but the Xolos were called offside regardless.
Then in the 57th minute, midfielder Cristian Pellerano, Hernan's brother, kicked a line drive from about 25 yards that caromed off a Galaxy player and sailed high toward the goal. Penedo jumped and got a hand on the ball, which then went over the net.
And in the 85th minute, Club Tijuana's Hercules Gomez headed a shot from just inside the six-yard box that was narrowly wide of the goal post.
"This is not the ideal result for us but it's not a bad result," Coach Cesar Farias said. "I think we can overturn this in Tijuana."
The Galaxy advanced to the quarterfinals by compiling a 3-1-0 record in the initial group-stage play last year against Isidro Metapan of El Salvador and CS Cartagines of Costa Rica.