The Cardinals moved a game in front of Pittsburgh in the National League Central after the Pirates fell to the Chicago Cubs. Cincinnati slipped 3 1/2 games behind in the division race.
Chance Ruffin (0-2) opened on the mound for Seattle in the 10th and struck out the first two batters. He then gave up a single to Kozma and Oliver Perez came on to replace Ruffin.
The first pitch from Perez escaped catcher Mike Zunino, allowing Kozma to score the winning run.
That gave the victory to Kevin Siegrist (3-1), who escaped a jam in the top of the 10th after putting two runners on with one out.