The only other expansion team to win their first three matches was the 2009 Seattle Sounders FC.
Roman Burki needed only two saves to keep his first clean sheet of the season against a San Jose team that had scored in each of its previous three games.
The Earthquakes (2-2-0, 6 points) remain winless on the road this season after a night that they finished considerably short-handed.
Gioacchini wasn't even the target on the play that he finished off to make it 1-0 in the 34th minute. Left-back John Nelson had come forward as part of an attacking move and was trying to pick out Klauss' late run through the middle.
The pass was a bit too far in front of the Brazilian striker, but Gioacchini outmuscled his defender to reach the loose ball and poke it past JT Marcinkowski.
It was 2-0 in first-half stoppage time on another play that didn't work out quite like intended. Klauss made a run forward and tried to connect a pass to Eduard Lowen that was deflected by a San Jose defender.
But Klauss ran onto it himself inside the penalty area and then stabbed a finish past Marcinkowski ahead of late-reacting defenders.
San Jose enjoyed their best period of play early in the second half, and Cade Cowell had two chances denied by Burki.
After that let off, Ostrak put the game out of reach in the 68th minute when his shot from the edge of the penalty area took a deflection off a San Jose defender that left Marcinkowski little chance.