Sebastian Lletget also scored in the first half for the Galaxy, who are off to a 4-1-0 start under new coach Greg Vanney.
Chicharito has scored in each of their four victories, and he slid between two Austin defenders to knock home a superb pass from Julian Araujo at close range in the 77th minute.
"It was very important to step up, because sometimes games like these are a tricky one,'' Hernandez said. "We prepared for this game and we didn't relax. We had to have our style and mentality, [but] we also kept a clean sheet. That was very important.''
New goalkeeper Jonathan Bond posted his first shutout for the Galaxy, while Austin was shut out in the Los Angeles area for the second time already. LAFC blanked Austin 2-0 in the franchise's inaugural game last month.
Chicharito also had a penalty saved by Austin's Brad Stuver in the first half of the first meeting between MLS' newest team and the league's record five-time champions. The Galaxy improved to 7-0-1 in Carson against expansion teams in their debut seasons since 2009.
"I thought there were some very good attacking moments in the game, probably more than we've seen this year,'' Vanney said. "I think we left three or four goals on the table. I thought the performance was a step forward, and as long as we're doing that, we're moving in the right direction.''
Only Seattle (13) has earned more points from the first five games of this season than the Galaxy (12), who have missed the MLS playoffs in three of the past four years. Vanney is having an immediate impact on this underachieving club -- from Chicharito's impressive play with better service up front, to the improved organization of a once-porous defense at the back.
"When you get a new management, you see all the ideas that they come in with, and I think it changed the energy around here,'' said Lletget, who has been with the Galaxy since 2015. "You hope for a great start, and we all felt like, 'Wow, this could be a special year.'
"We don't want to jinx it, but I think we all felt the shift. We have quality players, and I trust our staff. I've experienced some tough times with this club, and this is definitely the best I've felt about us.''
French forward Kevin Cabral made his Galaxy debut in the second half and appeared to score in the 85th minute, but he was ruled offside. The Galaxy signed the 21-year-old Cabral from Valenciennes in April as their third designated player.
Chicharito drew a penalty in the 26th minute when Jhohan Romana took him down in the box after a long run, but his fairly well-placed kick was saved by a diving Stuver to keep it scoreless.
The Galaxy showed superior creativity and aggression throughout the half, and they finally broke through in the 35th minute when Lletget ran onto a long pass from Daniel Steres at the same time Austin defender Hector Jimenez fell down with an apparently serious leg injury. Lletget beat Stuver for his first goal of the season.
Jimenez, a native of the Los Angeles suburbs who played three seasons for the Galaxy from 2011-13, left the field on a stretcher.