Page 194 of the PHB, under Making an Attack: "If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack."
That couldn't be any clearer. Attacks require attack rolls. It's that simple.
You're right, breath weapons don't break invisibility. They aren't attacks.
So if I use a fireball or something similar, it doesn't count as an attack? I can fire those and maintain invisibility? Pretty sweet.
I can live with Uncanny Dodge not working against things without an attack roll, if I can maintain invisibility using spells that don't require an attack roll.