Jeff Brown put the hosts ahead at the National Ice Centre before Colin Shields levelled at 12:14.
Ollie Betteridge restored Nottingham's lead six minutes later but Blair Riley hit back with a third-period equaliser for the Giants.
Overtime failed to separate the teams and Matt Carter converted in the shootout to win it for Nottingham.
Belfast missed the chance to go second in the standings and they are back in league action on Saturday night with an away game at Manchester Storm.
Belfast Giants suffered a shootout defeat in Friday night's Elite League clash against Nottingham Panthers.