Freese's blast in the second inning put the Cardinals ahead to stay and Beltran's shot in the sixth capped a four-run outburst that iced the contest for St. Louis.
Pete Kozma and Jon Jay each added RBI as the defending NL champs grabbed an early lead in the best-of-seven NLCS.
The Cards built a 6-0 lead and their bullpen took over after San Francisco posted a four-run fourth against starter Lance Lynn.
Edward Mujica (1-0), the third of six Cardinals relievers, claimed the victory with a 1-2-3 seventh and Jason Motte turned in a scoreless ninth for the save.
Giants starter Madison Bumgarner (0-1) was tagged for six runs on eight hits over just 3 2/3 innings.
Gregor Blanco delivered a two-run triple for San Francisco, which hosts Game 2 Monday.