Sean McMullen also had three RBIs for Greeneville (9-10).
Second baseman Casey Turgeon and shortstop Oscar Mercado were each 2-for-4 for the Cardinals (11-8), who managed two hits otherwise. Turgeon raised his average to .407.
Cardinals catcher Jose Godoy went for 1-for-3 and is hitting .408 on the season.
Agapito Barrios (1-0) allowed two runs in five innings for the victory.
Johnson City starter Steven Farinaro (1-3) allowed five runs in three innings, though four of the runs were unearned.
Cardinals reliever Kender Villegas allowed one run in three innings. He struck out three, walked none and gave up three hits.
Woodward was 3-for-4. McMullen was 2-for-4, as was right fielder Ramon Laureano. Astros leadoff batter Jason Martin was 2-for-5 with two runs and a triple.
Game two of the series is set for tonight at 7 in Greeneville.