David Bingham made five saves for the Galaxy (10-7-1, 31 points), who have dropped two of their last three MLS matches.
The Earthquakes struck early after Emmanuel Boateng's speed in transition resulted in a set piece. After three quick passes from the free kick, Qazaishvili made the most of the opportunity by unleashing a right-footed shot from outside the box that whipped past a diving Bingham to open the scoring in the 11th minute.
The 26-year-old Georgia international -- known as Vako -- has six goals in his last four matches spanning all competitions, including a pair of tallies on Wednesday as San Jose seized a 2-0 win over Houston.
The Earthquakes doubled their advantage in the 82nd minute after Salinas' right-footed shot from the right side of the box sailed to the bottom left corner.
San Jose capped the scoring three minutes later, as Salinas set up Thompson's left-footed strike that tucked into the bottom corner of the net.
Galaxy star Zlatan Ibrahimovic set up Dave Romney for a point-blank shot that was denied by Vega in the 38th minute. The 37-year-old Ibrahimovic, who leads the club with 11 goals, recorded a brace in a 3-3 draw against the Earthquakes at Stanford Stadium on June 30, 2018.
San Jose nearly doubled its lead in the 65th minute as Magnus Eriksson's through ball connected with Chris Wondolowski for an apparent goal. But the tally was disallowed as Wondolowski was ruled offside.