It is perhaps unsurprising that Tolkien, man of words that he was, invented a word to describe what goes on in this chapter and the end of the previous: eucatastrophe. It refers the moment when everything goes right, when the insurmountable odds are suddenly surmounted, and all becomes right with the world. It's not the same thing as deus ex machina, because in a true eucatastrophe Evil needs to contain the seeds of its own downfall - like how Sauron's lust for power blinded him to the possibility of its rejection.
(As an aside, I'll admit this here and here only: the Eagles showing up to save Frodo & Sam does feel like deus ex machina to me. I don't think Tolkien used them egregiously elsewhere, and I still think that the “‘Fly you fools’/why didn't they take the Eagles?” theory is relentlessly stupid. But yeah, they come out of nowhere here.)
This is the happy ending, the moment of joy that Sam was hoping for (as he notices, right before bursting into tears). It's also a chance to catch our breath after the last chapter.
And yet it is not the end of the story. Star Wars ended with Han, Luke, and Chewie getting their medals. If Tolkien had been writing it, the movie would have gone on to show the frenzied evacuation from the Rebel base on Yavin before a fleet of Star Destroyers could arrive. So we have plenty more to come.
Another aside: one of the most quintessentially Reddit-ish moments of my life happened while writing this post. I was trying to come up with a good example of a deus ex machina - which I didn't end up needing, but anyway - and I drew a blank, so I turned to Google. The top result was a thread on /r/Movies discussing deus ex machina, most of which consisted of people telling other redditors that they didn't really understand what deus ex machina meant, and the examples they were giving were just plot holes. Never change, Reddit.
The triumphant music continues to swell next time in the Steward and the King.