Kuroda (10-7) allowed only one run, struck out nine and issued a single walk while tossing 65 of his 107 pitches for strikes.
Rafael Soriano racked up his 25th save for American League East-leading New York, which was coming off a four-game sweep at the hands of the Oakland Athletics.
All-time Seattle hits leader Ichiro Suzuki, acquired by New York in a trade earlier in the day, went 1-for-4 in his Yankees debut.
Mariners starter Kevin Millwood (3-8) was touched for three runs on nine hits in a seven-inning stint.
John Jaso went 2-for-4 with an RBI for Seattle.