The 23-year-old Kershaw, a first-round draft pick (seventh overall) in 2006, has won eight of his last 10 starts and seven of eight since late June.
He improved to 15-5 on the season with eight innings of scoreless five-hit ball at Milwaukee in his start, lowering his earned run average on the season to a stingy 2.60 across 183 2/3 innings.
Kershaw won a previous career-high 13 games with the Dodgers last season and posted a 2.91 ERA across 204 1/3 innings and 32 starts.
The Dallas native has split four decisions in seven starts against the Cardinals.
Rhodes struck out his only batter, Andre Ethier, but Miles tripled to the right-center field wall off Fernando Salas (5-5) to tie the game at 1-1. Miles then scored when Rod Barajas, facing Jason Motte, grounded sharply to shortstop Rafael Furcal with the infield drawn in.
Furcal knocked the ball down, but his throw home was neither accurate nor on time.
Lance Berkman hit a leadoff homer in the second inning against Dodgers starter Nate Eovaldi, his 29th of the season, for the Cardinals' only run.
The Chico, Calif. native beat Pittsburgh, 7-2, in his last time out, allowing five hits and two runs in seven innings.
He's 2-2 in seven appearances - five starts - against the Dodgers with a 3.78 ERA.
St. Louis is 5-5 in his last 10 starts, including victories in the last four.
The Cardinals have won three of five from the Dodgers this season and have won seven of their last nine in the series.