Kuroda (11-8) took a no-hitter into the seventh inning when Elvis Andrus broke up the bid with a leadoff single. Michael Young added an eighth-inning hit but that was all the Japanese right-hander would allow.
Kuroda's shutout was his second of the season and fourth of his career. He yielded two walks and struck out five as the Yankees won their second straight in a four-game series pitting American League division leaders.
Swisher greeted him by sending a 3-2 pitch over the left-center field fence and Teixeira followed by lining a solo shot into the right-field stands for the AL East leaders.
Harrison had held the Yankees to five hits over 6 1/3 innings, walking three and striking out two for AL West-leading Texas.