The rule is the spell does exactly what it says in the spell description, no more no less. If there is no mention of a save then there isn't one.
All spell descriptions explicitly state whether the target must make a save and what happens to the target if they fail or make the save. If the description does not explicitly say the target of the spell makes a saving throw then they don't get one, they just take the damage or other effect.
An example of spell where you don't get a save is the Firebolt cantrip. The description does not say the target makes a save, so they don't. They just take the damage if they are hit:
Fire bolt (PHB p.242)
You hurl a mote of fire at a creature or object within range. Make a ranged spell attack against the target. On a hit, the target takes 1d10 fire damage. A flammable object hit by this spell ignites if it isn't being worn or carried.
A "similar" spell is Fireball. It automatically "hits" as there is no attack roll specified but the description states that the target does make a save and what the consequences are:
Fireball (PHB p.241)
A bright streak flashes from your pointing finger to a point you choose within range and then blossoms with a low roar into an explosion of flame. Each creature
in a 20-foot-radius sphere centered on that point must make a Dexterity saving throw. A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one.
An example of a spell where there is no attack roll and no save is Power Word Kill. If the target has 100 hit points or less it just dies:
Power Word Kill (PHB p.266)
You utter a word of power that can compel one creature you can see within range to die instantly. If the creature you choose has 100 hit points or fewer, it dies. Otherwise, the spell has no effect.