Rosar made his major league debut with the Yankees on April 29, 1939 at the age of 24. From 1939 to 1942, he served as the Yankees' back up catcher to the future Hall of Fame inductee Bill Dickey. By the middle of the 1940 season, Rosar was out-hitting Dickey with a .343 batting average compared to Dickey's .226 average, although he appeared in only half as many games as the Yankees were reluctant to relegate Dickey to second-string status. On July 19, 1940, he hit for the cycle in a game against the Cleveland Indians. Rosar appeared in 73 games in 1940 and set career-highs with a .298 batting average and a .357 on-base percentage. In 1941, he hit well above .300 until the final month of the season before tapering off to end the year with a .287 average in 67 games as, the Yankees won the American League pennant by 17 games over the Boston Red Sox. Rosar made only one appearance in the 1941 World Series as a late-inning defensive replacement for Dickey in Game 2 as, the Yankees went on to defeat the Brooklyn Dodgers in five games.