Galaxy dominated possession over the 90 minutes, but both teams had a number of opportunities to score.
Landon Donovan had the chance to put the Galaxy ahead from the spot in the 25th minute when Tally Hall brought down Jose Villarreal inside the area.
However, the goalkeeper redeemed himself to deny the former Everton loanee.
The win sees the Texan outfit join Sporting Kansas City, Montreal Impact and the New York Red Bulls at the top of the MLS Eastern Conference on 17 points.
Galaxy missed the chance to close the gap on Western Conference leaders FC Dallas to three points.