The lefty went 2-0, leading the Majors with 16 innings pitched and tied for the lowest ERA (0.00) among qualifying starters. Over the two outings, the defending NL Cy Young Award winner recorded 10 strikeouts while scattering three walks and 10 hits.
It is his second career weekly award, having previously won for the week of June 20-26, 2011.
Kershaw threw seven scoreless frames on Monday in a 3-1 win against the D-backs in Los Angeles, striking out six and allowing just four hits.
He was back at it on Saturday, leading the Dodgers to a 6-0 win against the defending World Series champion Cardinals at Dodger Stadium. Kershaw also doubled and scored a run as part of a four-run seventh inning. It was the first extra-base hit of his career.
Saturday's outing was Kershaw's first shutout of the season, as he struck out four without allowing a walk. He owns a scoreless-innings streak of 22 innings, which dates back to his May 8 start against the Giants.
Kershaw, who is 4-1 this year, currently ranks second in the NL with 61 2/3 innings pitched and is third among qualifying pitchers with a 1.90 ERA and 0.88 WHIP.