The Galaxy struggled with unfavorable officiating, but will get a second chance next week at home.
The best chance of the opening half fell to Herediano's Christian Montero, who headed off Carlo Cudicini's crossbar a few minutes before halftime.
The Galaxy almost gifted the home side the opening goal a few minutes after the restart. A miscommunication between A.J. DeLAGarza and Omar Gonzalez gave Herediano's Yendrick Ruiz a shot that went just wide. Mike Magee nearly punished Ruiz's profligacy immediately, hitting Robbie Keane's cross against the bar as the Galaxy went straight up the other end.
The Galaxy were right to feel aggrieved in the 68th minute, when a Mike Magee goal was ruled offside despite replays showing the forward's goal to have been clearly legitimate.
The crossbar came into play for a third time in the 87th minute. Herediano striker Ismael Castro went down ludicrously easy under minimal contact from Sean Franklin and drew a penalty, but pinged his effort off the frame of the goal to ensure parity remained.
The second leg of the quarterfinal matchup between Herediano and Los Angeles will be on March 13, at the Home Depot Center.