Mystra's Weave and the Multiverse
Using the Planescape parlance, this question is in the style of a "clueless prime". The Planescape setting is actually an over-setting, covering all of the other settings: even those that are isolationists like the Dark Sun or Ravenloft. Forgotten Realms is just one prime world amongst many others. It would be meaningless to assume that Mystra is responsible for the functioning of magic in the whole of the multiverse, let alone on, say Krynn or Mount Olympus or Oerth.
Each prime material world has its own space around it, called a crystal sphere. Toril is in Realmsspace, Krynn is in Krynnspace etc. Outside of the crystal spaces is some matter called phlogiston. A spelljamming ship can travel in a given crystal space (which allows people on Faerun to travel to Toril's moon Selune, for instance) or between crystal spaces (so you can go from Toril to Oerth, etc.). All of the magic in the Realmsspace is taken care of by Mystra. But beyond it arcane magic works by the rules of the particular sphere you go. (And it works without divine intervention in the phlogiston, otherwise spelljamming would be toast.)
Finally, spelljamming cannot get you beyond the prime material plane. If you go to other planes like Bytopia or Ysgard, etc. by means of a planeshifting spell, all those planes have their own rules and mechanisms for magic as well.
In short, no, Mystra and Toril's weave are not needed to cast spells beyond Realmsspace.
Generic Weave of 5e
There was no mention of the Weave as the source of all magic in the D&D multiverse before the 5e. It was a FR construct perceived as the embodiment of the goddess Mystra. Even in 5e, Ed Greenwood, the creator of the FR setting has tweeted:
The Weave is Mystra, and extends beyond Toril only so far as individual beings who are Weavemasters do (example: the Chosen, and Mystra herself when she went into the Hells), and even they can't access the full Weave when they're outside Realmspace.
Yet, the 5e Player's Handbook now identifies the Weave as the source of magic everywhere. This generalisation might have been introduced in order to facilitate the return to Vancian spellcasting in 5e after the 4e. If you think about it, declaring the Weave as the source of magic everywhere is quite unnecessary. It explains something supernatural, magic, with some other supernatural thing, Weave.
Imagine we had no FR, removing the discussion of the Weave from the Player's Handbook would have almost zero impact on D&D. Just for FR, it was part of the whole lore around the goddess Mystra, and no such baggage needs to be carried to the other settings.
Magic without the Weave in Forgotten Realms
About three years after the first version of this answer was written, Ed Greenwood was asked about his personal view of the Weave on twitter (March 9, 2020). It is an interesting read, confirming the main points of the answer and describing how he does not see the Weave as the only conduit to magic even in the Forgotten Realms:
Toril and all other worlds have multiple ways of accessing the energies of each world. And usually call the non-mechanical, non-engineering ways of accessing those energies "magic." ONE of the ways on Toril is arcane magic ... The Weave IS Mystra, the goddess of magic, and the Weave only extends throughout Realmspace (Toril and moons and the void/heavens around them). Other worlds have other ways of accessing energies, and may have their own equivalents of the Weave (not identical, and not governed by Mystra) or may not. ... Other worlds have other systems of magic, and even Toril has other systems of magic (divine magic, for example).
Inquired further about "other" systems of magic, he continues:
Blood magic (censored out of the published Realms due to TSR's fears that there'd be gory real-world experiments, and lawsuits), candle magic, table magic, and more. I deliberately wanted magic to be so vast and varied that even "rules-lawyer" players couldn't remember it all at the gaming table, which would lead to a better roleplaying experience, as opposed to metagaming.