Japanese right-hander Kuroda (8-6) allowed six hits and struck out five to help the Yankees win their third straight game and improve to 51-42 for the season.
Boone Logan recorded three clutch seventh-inning strikeouts to preserve the lead and Mariano Rivera finished off the eight-hitter for his 30th save on a soggy, windy evening at Yankee Stadium.
Gardner and Cano accounted for New York's offense with a pair of fifth-inning RBI singles after the home team had failed to score against Twins' starter Scott Diamond before rain interrupted proceedings for 73 minutes.
Joe Mauer, Brian Dozier and Trevor Plouffe each had two hits for struggling Minnesota, who slipped to their sixth successive defeat, and their 12th in 13 games.
In Atlanta, the Braves suffered a costly evening game at Turner Field where they were beaten 4-2 by the Cincinnati Reds and also lost two more outfielders to injury - the Upton brothers, B.J. and Justin.
Cincinnati starter Bronson Arroyo (8-7) held a depleted Atlanta to just three hits in seven strong innings while Brandon Phillips hit a two-run single in the first as the visitors evened the four-game series at one game each.
Sam LeCure pitched a perfect eighth and Aroldis Chapman yielded a run in the ninth before earning his 21st save as the Reds improved to 52-41. The National League East leading Braves slipped to 53-40.
Atlanta, without right fielder Jason Heyward who strained his right hamstring the previous day, lost center fielder B.J. Upton (right adductor muscle strain) in the first inning and right fielder Justin Upton (left calf strain) in the seventh.
In Detroit, Doug Fister (7-5) threw six strong innings after his batters had supported him with seven early runs to propel the hosting Tigers to a 7-2 victory over the Texas Rangers in front of a capacity crowd of 41,686 at Comerica Park.
Austin Jackson and Torii Hunter hit back-to-back doubles to start the game, and Jhonny Peralta and Victor Martinez drove in two runs apiece as Rangers starter Justin Grimm (7-7) was removed due to a sore right forearm with one out in the fourth.
Martinez extended his hitting streak to 14 games, batting .446 during that stretch with 10 runs scored and nine RBIs, for the Tigers (51-41) to remain 2-1/2 games ahead of the second-placed Cleveland Indians in the American League Central.
In Miami, Washington starter Stephen Strasburg (5-7) made an early exit in one of the worst games of his career as the Nationals were hammered 8-3 by the Marlins.
Strasburg walked three of the first four batters he faced and was removed after allowing seven runs over two innings at Marlin Park, where Marcell Ozuna drove in three runs and Giancarlo Stanton homered for Miami.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Peter Rutherford)