"His last outing was pretty good, so let's not get too far ahead of ourselves," Matheny said. "He's got to get back to work."
Ellsbury had an RBI single in the third as five straight Yankees reached safely, and added a two-run single in the fourth. Murphy had a two-run single in the fourth.
The Cardinals stranded two in the second and left the bases loaded in the third when Molina popped out to second. Kuroda has allowed just one extra-base hit with the bases loaded in 65 career plate appearances, limiting opponents to 14 hits in 57 at-bats with a double.
St. Louis chipped away with single runs in the fourth, fifth and sixth and Wong's RBI single off David Robertson in the eighth cut the deficit to three. The first two Cardinals reached in the ninth before Robertson struck out the next three, including pinch-hitter Matt Adams, to earn his 12th save in 13 chances.
NOTES: All three games were sellouts. Paid attendance of 45,267 in the finale was the Cardinals' 13th in 24 home dates. ... Adams did not start at 1B due to calf tightness and Matheny wasn't sure whether he'd be able to start Thursday. ... The Yankees have a day off Thursday before beginning a seven-game homestand against the Twins, Mariners and A's. Vidal Nuno (1-1, 5.49) starts Friday against Minnesota's Ricky Nolasco (2-5, 6.12). The Cardinals open a four-game series at home against the Giants Thursday, with Jaime Garcia (1-0, 4.26) opposing Ryan Vogelsong (3-2, 3.20). ... Matt Holliday singled in the third and has reached base safely in all 24 home games. ... The Yankees are 6-3 overall against St. Louis.