A left-handed batter, Kubek signed his first professional contract with the Yankees and rose rapidly through the team's farm system. He was 21 years old when he played his first game in Major League Baseball in 1957, and—except for one year (1962) spent largely in the U.S. military—remained with the Yankees until his retirement due to a back injury at the close of the 1965 season. In his prime he formed a top double play combination with second baseman (and roommate) Bobby Richardson on an infield that also featured third baseman Clete Boyer.
In 1957, teammate Sal Maglie praised Kubek when talking to Robert Creamer of Sports Illustrated. "He's a nice boy, that Kubek. He's going to be a great player. He's a fine hitter right now. He hits with the pitch and it's hard to fool him. His wrists are so quick he can wait to see where the pitch is thrown before he commits himself, and then he can push it to left or pull it to right. And he's never satisfied with himself. That's the sign of a good ballplayer."
In 1957, Kubek won the American League Rookie of the Year Award. In Game 3 of the 1957 World Series, he had one of the best World Series games a rookie has ever had, going 3 for 5 with two home runs, three runs scored, and four RBI. Kubek is one of five rookies to hit two home runs in a World Series game. Another Yankee, Charlie Keller, had performed the feat in the 1939 World Series against the Cincinnati Reds and St. Louis Cardinal Willie McGee homered twice in the 1982 World Series against the Milwaukee Brewers. All three of these feats occurred in a Game 3; Kubek's and McGee's both occurred at Milwaukee County Stadium. The Atlanta Braves' Andruw Jones homered twice in Game 1 of the 1996 World Series at Yankee Stadium. Michael Conforto of the New York Mets became the fifth member of this club when he hit two in Game 4 of the 2015 World Series on October 31, 2015.