Juan Garcia, 35, was pronounced dead at 6:26 pm local time, a prison system spokeswoman told AFP. He was put to death by lethal injection.
Garcia, who was a teenager at the time of the murder, previously had sought clemency over mental health issues, but filed no appeal yesterday.
Prison officials said that Garcia's final remarks were, in part: "The harm that I did to your dad and husband, I hope this brings you closure to all of you. I never wanted to hurt any of you."
US states using the death penalty have faced obstacles over shortages of lethal injection drugs after European suppliers stopped supplying pentobarbital for use in executions.
Although Texas was affected by the shortage in the past, it now secures its pentobarbital from a compounding pharmacy.
The shortages have prompted prison departments in the states that still allow the death penalty to seek new supply sources or new drug mixes.