According to IGN, A Link to the Past was released before Link's Awakening

Link's Awakening:

US: December 31, 1993
Japan: June 6, 1993
Europe: Released
Australia: Released
A Link to the Past

US: November 21, 1992
Japan: Released
Europe: Released
Australia: Released
Source:
IGN: The Legend of Zelda: A Link to the Past
IGN: The Legend of Zelda: Link's Awakening