Goalkeeper Andrew Tarbell made five saves as San Jose halted a 12-match winless stretch. The Earthquakes were 0-7-5 during their drought.
Midfielder Roland Lamah scored for FC Dallas, who lost at home for the first time this season. They are 7-1-4 on the home turf.
The Earthquakes took a 2-1 lead in the second minute of the second half when Hyka scored his second goal of the season.
Midfielder Aníbal Godoy delivered a sharp pass to Hyka, who sent a squib-type shot that deflected off defender Reggie Cannon. Goalkeeper Jesse Gonzalez was unable to stop the ball as it caromed off him and into the net.
Meanwhile, Tarbell protected the one-goal lead against a major assault over the next 41 minutes as he thwarted forward Michael Barrios on three occasions and had good fortune when a rocket blast by forward Maximiliano Urruti hit the crossbar and caromed out of play.
Lamah sent a header over the net in the 75th minute and forward Cristian Colman mishit a wide-open header to the left in the 80th minute.
San Jose finally created a cushion in the 88th minute when forward Chris Wondolowski dribbled along the end line and threaded a superb pass to Qazaishvili, who sent a sliding right-footed shot into the net for his sixth goal of the season.
The late goal assured that the Earthquakes would be victorious for the first time since defeating Minnesota United on May 12.
San Jose took a 1-0 lead in the 19th minute when Qazaishvili eluded Gonzalez, who came all the way out to the front left corner of the box. Qazaishvili also meandered around FC Dallas defender Matt Hedges, leaving him with a wide-open net as he sent a right-footed shot into the goal.
FC Dallas answered four minutes later as Cannon stole the ball and fed Barrios in the right side of the box. Barrios delivered a pass to Lamah, who easily executed the left-footed tap-in from in front of the net for his eighth goal of the campaign.
FC Dallas took 18 second-half shots to finish with 23. The Earthquakes took 10 of their 17 shots in the first half.
FC Dallas 1-3 San Jose: West's worst beat West's best
San Jose shocked Western Conference leaders FC Dallas, winning just their third game of the season in comfortable fashion.