The sheriff's office said the suspect was "down" and had been taken to hospital.
Sheriffs then tasered the suspect and opened fire when tasering couldn't contain him.
Elizarraras was taken to Scripps Memorial Hospital in La Jolla and as of Monday, was listed in stable condition and is expected to survive.
The concert, for which tickets were sold out, was scheduled to follow a full slate of horse races at the track, the station reported. No one was hit, but San Diego Sheriff's deputies returned fire and Elizarraras went down. After an argument ensued, Elizarraras pulled out a silver plated semi-automatic handgun and reportedly fired three shots in the air.
No other details were immediately known.
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Bournemouth had defended stoutly throughout and could easily have scored themselves before Pedro's shot deflected into the net 18 minutes from time .
The gunfire happened as the final race of the day was underway, the San Diego Union-Tribune reported.
The exchange of gunfire occurred just before Ice Cube was set to take the stage as part of Del Mar's Summer Concert Series, his performance set to be the last show of the summer schedule.
The Del Mar Thoroughbred Club said in a statement that under its special events protocol, substantial law enforcement and private security were on site during the event.
At 7:22 p.m., the Del Mar Racetrack tweeted that the "situation has been contained" and said the concert "is moving forward as planned".
Ambulances and helicopters were also seen shortly after the shooting.