Eh, the Eastern Roman Senate ceased existing sometime in the 13 or 14th century, while the Eastern Roman Empire lasted up to 1453, so if we take a bit of a more expansive view on what Rome is than "the western half because that's the half closer to western historians" the senate fell first.
The Roman Senate was more of a social club for at least few hundreds years prior to that. You could argue it hardly existed before the fall, except in name (losing mostly all authority when Rome transitioned from a republic to an empire).
It's pretty amusing to imagine a future with a failed-US, and how existing institutions might outlive the country. I wonder if there's been any good books using that idea?