Hernández took a pass from Dejan Joveljic and scored in the 54th minute, sending a right-footed shot into the upper left corner to give the Galaxy the lead. Puig added an insurance goal 10 minutes later off an assist from Mark Delgado to the left wing that Puig gathered and slotted into the far right corner.
The Galaxy (1-4-3) beat Austin for a third consecutive time and improved to 4-1-0 all time in the series.
Austin (2-4-2) has gone 0-3-2 in its last five matches, with only one goal scored. The team has scored just six goals this season. It scored six goals in the first 112 minutes of last season.
The Galaxy had advantages of 18-11 in shots and 8-2 in shots on goal.
Jonathan Klinsmann saved two shots to earn the clean sheet for the Galaxy. Brad Stuver made six saves for Austin.
Austin returns home to face the San Jose Earthquakes next Saturday. The Galaxy travel to play Orlando City next Saturday.