Matt Carpenter added a two-run homer on a four-hit day. Adams homered for the second consecutive game and added an RBI double and walk.
The Cardinals have dominated the Reds at home the last decade, losing just three of 28 series with two splits.
Westbrook (1-1) threw his fourth career shutout, walking four and striking out three.
Westbrook pitched his first shutout since Aug. 9, 2006, against the Angels when he was with Cleveland, and this was his 15th career complete game. He held the Reds hitless in six at-bats with runners in scoring position, and contributed his 11th career RBI with a single in the seventh for St. Louis' final run.
Bailey (1-1) had gone 23 consecutive shutout innings — including his no-hitter against Pittsburgh last season — before the Cardinals broke open a scoreless game with four runs with two outs in the fifth.
Bailey was charged with seven runs in five-plus innings and dropped to 3-8 with a 5.52 ERA against the Cardinals, including 0-4 with a 6.93 ERA at Busch Stadium.
The Cardinals advanced only two runners into scoring position in the first four innings and Bailey got two quick outs to start the fifth before running into trouble against the top of the lineup. St. Louis scored 10 runs on 11 hits in the fifth, sixth and seventh off Bailey, Logan Ondrusek and Alfredo Simon.
Jay's second homer just cleared center fielder Shin-soo Choo's leaping attempt at the wall. Carpenter followed with a single and Matt Holliday hit an RBI double, prompting a visit from pitching coach Bryan Price.
Beltran then hit his first homer of the season on to make it 4-0.
Adams hit a two-run homer in the sixth off Bailey. Carpenter's two-run shot homer three batters later made it 8-0.
Notes: Jay has 19 career homers, six of them against the Reds. ... Choo was 0 for 2 with a walk and hit by pitch to end a four-game streak of two hits each game. ... Cincinnati's Todd Frazier was 0 for 11 in the series after opening the season with a six-game RBI streak.
Link to box score: St. Louis 10, Cincinnati 0