In 1993–94, playing full-time under new coach Pierre Maguire, Cunneyworth appeared in 63 games with Hartford, scoring nine goals and 17 points. On March 11, 1994, with the Whalers well out of the playoff hunt, Whalers GM Paul Holmgren traded Cunneyworth, Gary Suter and the Whalers third round pick in the 1995 NHL Entry Draft to the Chicago Blackhawks for Frantisek Kucera and Jocelyn Lemieux.