Last night was our first use of the haste spell in 5e. One turn, the bard cast shatter with a second-level spell slot, then used his haste action to use a magic staff which cast thunderwave.
I'm not sure if this violates the "only one non-cantrip spell per turn" rule. On the one hand, the haste action says you can "use an object", and a staff is an object. On the other hand, thunderwave is a non-cantrip spell.
The staff in question is a custom magic item, but in this question I'm more interested in the Rules As Written interaction between items-that-let-you-cast-spells and haste "Use An Object" actions. Let's use the "Staff Of Thunder And Lightning" as an analog:
You can use an action to cause the staff to issue a deafening thunderclap, audible out to 600 feet. Each creature within 60 feet of you (not including you) must make a DC 17 Constitution saving throw. On a failed save, a creature takes 2d6 thunder damage and becomes Deafened for 1 minute. On a successful save, a creature takes half damage and isn't Deafened.