The incident happened just before 11 a.m. on the 1900 block of East 105th Street.
Police say the 22-year-old suspect started attacking his mother. The stepfather was stabbed when he tried to defend his wife, according to investigators.
The stepfather was taken to St. Francis Medical Center in Lynwood, where he was pronounced dead. The victim, a 54-year-old mechanic, has not been publicly identified.
After the stabbing, the suspect fled the home but was later spotted by officers.
"When they were about three blocks away from arriving at the location, they noticed an adult male Hispanic, who was shirtless and part of his torso had blood on it. He did not appear to be injured. He was acting peculiar," said Lt. Jeff Nolte with the LAPD. "Officers made contact. There was a short chase. He was taken into custody without incident."
Authorities said the suspect is a known gang member. The motive for the crime remains under investigation. Witnesses said the attack was unprovoked.