After exercising his right to demand a trade because of lack of playing time, he was traded along with Henry Cotto, Rich Bordi and Porfi Altamirano from the Cubs to the Yankees for Ray Fontenot and Brian Dayett at the Winter Meetings on December 4, 1984. While with the Oakland Athletics he was known for almost exclusively catching all of starting pitcher Bob Welch's games, most notably during the 1990 season in which Welch won 27 games and earned the Cy Young Award trophy. On July 28, 1991, he caught Dennis Martínez's perfect game against the Los Angeles Dodgers.
He had the highest average among all catchers in the 1980 baseball season with a .318 batting average. On May 15, 1981, Hassey caught starting pitcher Len Barker's perfect game against the Toronto Blue Jays. On June 13, 1984, Hassey was involved in a high-profile trade, when the Cleveland Indians traded him along with Rick Sutcliffe and George Frazier to the Cubs in exchange for Joe Carter, Mel Hall and Don Schulze.