Carpenter drove in Pete Kozma with a one-out single. Jay followed with an RBI triple and Adams added a run-scoring single.
St. Louis, which leads Pittsburgh by 2½ games in the NL Central, won five of six games on its homestand.
"You show up every day wanting to play and you end up with good results," Carpenter said. "This was a big day for us."
Erik Kratz drove in the Phillies' run with a single in the fourth.
Philadelphia outfielder Steve Susdorf grounded into a double play in the seventh in his major league debut. He was recalled earlier in the day to replace Domonic Brown, who was placed on the seven-day concussion disabled list.
The Phillies, who fell to 9-19 against NL Central foes, remain eight games behind Atlanta in the NL East.
"At times, we have trouble putting anything on the board," Manuel said. "We got some hits, but we didn't have any real good chances."
Notes: St. Louis OF Matt Holliday is expected to come off the disabled list on Saturday. Holliday's wife Leslee gave birth to a son, Reid Joshua, on Wednesday night. Holliday was placed on the 15-day disabled list on July 12 with a right hamstring strain. ... The Cardinals open their lengthy road trip with three in Atlanta. Adam Wainwright (13-5, 2.44) will face Mike Minor (9-5, 2.88) in the opener of a three-game series. Philadelphia begins a three-game series in Detroit on Friday. Cole Hamels (4-12, 4.16) takes on Doug Fister (8-5, 3.90) in the series opener. ... Allen Craig had a nine-game hitting streak snapped Thursday. ... Jimmy Rollins and John Mayberry Jr. each had two hits.