Christopher Young was declared dead on July 17, 2018 at 6:38 pm local time, the southern state's Department of Criminal Justice said.
Mitesh Patel, the victim's son, had campaigned in recent weeks to ask the Texas governor to spare the prisoner, a father of three.
Mitesh Patel visited Christopher Young on Monday and described the meeting as emotionally moving for both of them.
Texas to execute 'ice pick killer' for rape and murder
The last words of the condemned man were addressed to the family of the victim: "I want to make sure the Patel family knows I love them like they love me."
"Make sure the kids in the world know I'm being executed and those kids I've been mentoring keep this fight going," he said.
In November 2004, while under the influence of alcohol and narcotics, Young shot the manager of a San Antonio supermarket.
He was also charged with sexual assault of a woman that same day.
Young, who was a member of a violent gang, said he had no intention of taking the life of the 50-year-old businessman named Hasmukh Patel.
The scene was filmed by a surveillance camera.
One of Young's daughters pleaded with the Texas authorities not to execute her father.
But the state's pardon and parole board ruled out any leniency.
Texas is the state that executes the most inmates in the United States: Christopher Young is the eighth sentenced to die since the beginning of the year.
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