5E Realms lore does not specifically address this. 4E Realms lore, on the other hand, does. Their exact origin is subject to some debate, even among the elves.
Collectively, the elves are known as the Tel’Quessir6 ("The People"), a title that encompasses Eladrin, Wood Elves, High Elves, Sea Elves, and so on. It is 'widely accepted' that the Tel’Quessir are native to the Feywild1. It is believed that they first immigrated to Abeir-Toril over a hundred millenia ago, while the exact date is uncertain it is believed to have been prior to −24,000 DR, with some manuscripts indicating it was as early as −30,000 DR, the same time in which dragons were setting up their empires (Note: Source on this is from AD&D)4. The Wild Elves were the first to arrive.1
The exact origin of the Elvish race is subject to some debate. Some ancient manuscripts suggest that they were the result of a battle between Gruumsh and Correlon that occurred in the Feywild. During said battle, Correlon was injured and his shed blood became the Eladrin.2. Other types of elf, not liking the superiority this implies in that the Eladrin are the first and 'purest' of elves, disagree with this theory3. Another theory postulated is that since the Feywild tends to be a 'reflection' of the Prime Material Plane, and thus produces creatures that are echoes of creatures from the Prime, it is also possible that the Elves are one such echo... though what, exactly, they would be an echo of is uncertain1.
Perhaps also worthy of mentioning is the current in-lore explanation for how the Feywild (also known as Faerie) is treated according to current lore. The Feywild was created as it exists today, a 'lighter' echo of the Prime Material Plane. It was made by Primordials in the far distant past, before recorded history begins.5
For a while, the Feywild existed alongside the Prime and interaction between the two was common. But, at some point in ancient history, the two drifted apart and travel between them became increasingly difficult2. This is the explanation for why, in prior editions of D&D, the Feywild didn't "exist" in the Forgotten Realms. It was out there, you just couldn't get to it.
The Spellplague (the event that 'kicks off' 4E in the Realms) changed that, reshuffling the cosmology to drag the Feywild back into close proximity, and placing it in the accessible location it resides in today5.
Checking through published 5E Realms material, it does not contradict any of this... the 5E Realms cosmology is essentially the same as the 4E Realms cosmology, and the lore established within 4E is considered to hold true in 5E, except where directly overridden. And even then, the changes are usually justified in-lore (such as the array of gods and goddesses getting rejiggered due to Ao meddling with them).
The primary lore sources we have in 5E Realms are the Sword Coast Adventurer's Guide, the DMG, and published adventures. None of these go into detail on the origins of the Elves, or even on the specific nature of the Feywild... thus we may assume the lore of older sources hold true.
Sources
1 Brian R. James and Ed Greenwood (September, 2007). The Grand History of the Realms. (Wizards of the Coast)
2Bruce R. Cordell, Ed Greenwood, Chris Sims (August 2008). Forgotten Realms Campaign Guide. (Wizards of the Coast)
3James Wyatt (December 2007). Dragon 361: A Fractured Family. Wizards of the Coast.
4Steven E. Schend and Kevin Melka (1998). Cormanthyr: Empire of the Elves. (TSR, Inc) NOTE: AD&D source
5Dungeon Master's Guide 4th edition. (Wizards of the Coast)
6Kim Mohan ed. (2015). Sword Coast Adventurer's Guide. (Wizards of the Coast)
I think the Arcane Age and Karsus's Fall entered canon sometime around 1995-1996.
Karsus was described as a "lesser-known" deity in a half-page article by Eric Boyd published in the February 1995 issue (#104) of the Polyhedron magazine. At this point, Karsus is not a dead power and his demise (from greater deity to demigod) was attributed to his poisoning by a Netherese mage named Wulgreth, who sought to steal immortality from Karsus (ironic, isn't it?). So we can conclude that early 1995, Karsus's story as we know it was not yet canon.
TSR released the Netheril: Empire of Magic (N:EoM) boxed set in November 1996. Yet we can argue that Netheril storyline was already entering the FR canon before N:EoM's publication based on evidence from the sourcebook Faiths and Avatars (F&A), which was published in March of that year. F&A has plenty of information about Mystra that foretells the then-upcoming Arcane Age storyline: Mystryl is given as one of Mystra's aliases; her speciality priests can also be of the CN alignment (Mystrl was CN); one of Mystra's symbols is none other than Mystrl's symbol, a single blue-white star. It is worth noting that one of the authors of F&A was Eric Boyd.
Powers and Pantheons (P&P), a companion to and continuation of F&A, was published in 1997. At this point it is clear that the whole Karsus story is canon: P&P lists Karsus as a dead demigod, describing explicitly the events in the earlier Netheril material; including his 12th level spell, and how he used it to steal Mystrl's power and ended up dead.
1998 saw the expansion of the Arcane Age material further with the publication of Cormanthyr: Empire of Elves. This product is also in full support of the Karsus's Fall story, including all the events in the earlier Arcane Age material.
The conclusion is that Karsus's Fall became canon during the 2e of AD&D.
Best Answer
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:
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:
Inquired further about "other" systems of magic, he continues: