With the score tied 2-2 in the 12th, Mike Carpenter drew a leadoff walk from John Ely (0-2) and went to second on Pete Kozma's sacrifice.
One out later, Jay lined a double into the right field corner, chasing home Carpenter with the go-ahead run.
The Cardinals' win put them one game ahead for Los Angeles in the wild card chase.
Shelby Miller (1-0) collected his first major league win with a scoreless 11th and Jason Motte nailed down his 35th save for St. Louis.
Andre Ethier supplied the Dodgers' offense with a two-run homer.
Notable deaths of 2014 [PHOTOS]