Garza was the 1981 Indianapolis 500 Rookie of the Year after starting sixth, leading 13 laps, and finishing 23rd at the age of 19.

At 19 years, two months, and nine days, Garza is the second youngest driver to compete in the Indianapolis 500 and was not surpassed until 2003 by A. J. Foyt IV.

In 1986, he tested a Brabham Formula One car, during free practice of the 1986 Mexican Grand Prix.

After he finished his career in CART, he tested with Minardi in F1 during 1987-88 but never got the required sponsorship to be awarded a race seat and he never made his debut in F1.

He came back to Mexico and later he became a team owner in Mexican F2 and F3000 and sometimes raced when his drivers were not up to standard, or absent for some reason.