The 37-year-old right-hander has struck out six and thrown 72 pitches.
St. Louis starter John Gant didn't allow a hit until Ronald Acuña Jr. singled to open the fourth. Freddie Freeman followed with a single, but the inning ended when Austin Riley lined out to third.
Morton entered 2-12 with a 5.68 ERA in 18 career starts against St. Louis.
Plagued often by big innings this year, Morton has been steady with a fastball up to 96 mph, mixing in plenty of curves and cutters, too.
The Braves led 2-0 on Guillermo Heredia's third homer in the fifth and Ozzie Albies' RBI triple in the sixth.