Lynn (8-1) surrendered just one run on five hits, walking one and striking out six while tossing 74-of-105 pitches for strikes.
Beltran (3-for-5) delivered an RBI single in the fourth inning for a 4-0 St. Louis lead and later capped the scoring with a two-run shot in the sixth.
Yadier Molina, playing while appealing a one-game suspension for making contact with an umpire, added a solo homer and a run-scoring single as the National League Central-leading Cards took the opener of a four-game series.
Matt Carpenter added three hits and scored all three times in the victory.
D-backs starter Trevor Cahill (3-6) was touched for five runs on nine hits in a five-inning effort.
Paul Goldschmidt knocked in the only run for NL West-leading Arizona.