The incident began about 10:30 a.m. when Carlos Hernandez Romo found his ex-wife with another man and cut them with a chain saw, said sheriff's Deputy Scott Gage.
The boyfriend fled the apartment naked and asked people to call police, a neighbor told KABC-TV.
The woman also came rushing out of the burning apartment trying to extinguish her flaming clothes, Gage said.
It was not immediately known how the fire started.
The woman was rushed to a nearby hospital with cuts to her upper torso and burns to 90 percent of her body, Gage said. She was listed in critical condition.
Her boyfriend suffered cuts to his upper body and was taken to a nearby hospital where he was treated and released.
The identities and ages of the victims were not immediately available.
Romo was booked for investigation of assault with a deadly weapon, Gage said.
Lennox is located about 10 miles southwest of downtown Los Angeles.
