The spell needs a save DC; the DC of the Sorcerer who triggers it is used
In the case you offer, fireball, the magical effects require that anyone within the fireball's radius make a saving throw.
What is a saving throw made against? The spell's DC.
Bards, paladins, sorcerers, and warlocks use Charisma as their spellcasting ability, which helps determine the saving throw DCs of spells they cast. (Basic Rules p. 62)
The Difficulty Class for a saving throw is determined by the effect that causes it. For example, the DC for a saving throw allowed by a spell is determined by the caster’s spellcasting ability and proficiency bonus. (Basic Rules p. 62)
How do you set the DC for a spell from a wild magic surge?
Use the DC of whomever triggered the surge. The Wild Magic Surge rules text (PHB p. 103) does not offer a separate DC rubric for that particular magical effect, so it is tied to the sorcerer who is the cause of that magical effect.
Is that stated explicitly anywhere in the Rules? Not as far as I can find.
Does it make sense? Yes.
Why? D&D 5e design principles were to make the game simpler, not more complex. If you accept that the "you cast a spell" rules text is a pointer to how the DC for a spell effect from a Wild Magic surge is determined (so that the DM does not have one more thing to rule on or figure out) it meets the simplicity standard.
Try the common sense test
Should a wild magic surge from a 19th level sorcerer be more potent than from a 1st level sorcerer? (Difference in DC being +6 to +2 based on proficiency alone) and thus harder to save against?
Yes, common sense would suggest that this is the case.
(If I can find a Crawford tweet on this, I'll add it).
From the Sage Advice Compendium:
Does a sorcerer’s Wild Magic Surge effect replace the effect of the spell that triggered it, or do both effects happen? The spell and the Wild Magic Surge effect both happen.
This supports the point that the two are directly tied to the Sorcerer who cast the original spell that triggered the wild magic surge. It shows a linkage.
Further linkage between the Sorcerer and the wild magic spell effect
A wizard multiclasses into Wild Magic sorcerer. Do spells cast from their spellbook trigger Wild Magic Surge if they are on the sorcerer spell list or do they have to gain them from Sorcerer to trigger? From the multiclassing rules: “Each spell you know and prepare is associated with one of your classes” (PH, 164). This rule means only the spells gained from levels in sorcerer trigger Wild Magic Surge.
Direct linkage between sorcerer, sorcerer spells, and Wild Magic Surge.
Your sorcerer casts a sorcerer spell, a wild magic effect (in this case a fireball spell) goes off at the same time thanks to the roll on the table, so the effect is that your sorcerer cast both spells thanks to the wonders of wild magic.