Salvador Acevedo, 45, was the father of the two slain children, 5-year-old fraternal twins, Los Angeles County sheriff's deputies said.
The shootings occurred just before 9 p.m. Saturday in an SUV parked in the driveway of a home in the 1100 block of South Hicks Avenue.
Authorities said Acevedo was driving with his girlfriend, Iris Oseguera, their young twins and three other teenage children when the two adults began to argue.
Acevedo pulled out a handgun and shot Oseguera, 33, and the twins, striking each victim in the upper torso.
Authorities said Acevedo then drove the teens to a nearby relative's house before turning the gun on himself.
The relationship to the three teens who witnessed the murders was not immediately known.
The man and the woman were found dead at the scene. The two children were rushed to Los Angeles County-USC Medical Center where they were pronounced dead.
Family and friends said the couple had a rocky and sometimes violent relationship. They said Acevedo was arrested five months ago for domestic violence for attacking Oseguera.
"A couple of weeks ago there was an outright brawl between the two of them, very violent, very loud," said Jackie Miller, a family acquaintance.
Candles and flowers were left at the makeshift memorial Sunday where the shooting took place.
"I don't believe it yet," said Rigoberto Carrillo, a family friend. "Everybody, we are confused."