The result in the LA rivalry dubbed "El Trafico" was a reversal of LAFC's 6-2 thumping of the Galaxy at the MLS is Back tournament in Orlando, Florida, last month.
The Southern California native was unmarked in the box when Julian Araujo's cross found him and he nodded a header behind Kenneth Vermeer.
Lletget doubled the lead in the 54th, running unmarked into the penalty area and finishing off an Araujo cross.
Four minutes later, Vela limped off with an apparent knee injury.
The league's reigning Most Valuable Player was playing his first match in more than five months, having opted out of the MLS is Back tournament to stay with his pregnant wife.
Vela was brought up short by the non-contact injury as he received a short pass while cruising down the flank.
"We'll need at least a day or two to fully assess, but just before that play he took a shot, the shot was blocked," LAFC coach Bob Bradley said, adding that it could be a medial collateral ligament injury.
"That is usually an MCL-type situation so that's what we will look at in the next day or two."