Paulino (2-1), who was coming off seven shutout frames in Wednesday's start against Baltimore, continued his hot streak at Yankee Stadium, scattering six hits while walking two.
Tim Collins and Aaron Crow finished up with a combined 2 1/3 scoreless innings for the Royals, who took the opener of a three-game series.
Mike Moustakas and Jeff Francoeur each socked two-run homers and Eric Hosman drove in a pair with a double to provide Kansas City's offense.
New York starter Hiroki Kuroda (3-6) lost his second straight, surrendering three runs on seven hits in a 5 1/3-inning effort.
Celebrity Families of 2014 [PHOTOS]