I initially thought you wanted to know the origin in pop music as a seasonal ending, as transcription questions are off topic; so, to cover both…
It's based on Pachabel's Canon originally… but it's kinda been jazzed up over the years. See Wikipedia - Jingle Bells, Melody
If you want to know the term for this 'borrowing' of one piece within another, it's generally called a quote. A quote doesn't have to follow the chord structure of the original, so long as the melody is recognisable.
To its popularity, if not the actual origin [see later]
Two words - Bruce Springsteen.
He has been playing Santa Claus is Coming to Town live at xmas since the mid 70's, eventually releasing one live recording as a single in 1982.
It is played a million times a year, everybody knows it… & it closes with the line from Jingle Bells, slowed down.
It is therefore now a part of common culture.
I'm not saying he started it, but he sure as heck popularised it.
Here's a version from 1978, earliest I could find…
..and from comments, we have an earlier usage, by maybe only a year or so in 1974, but here's Mud's Lonely This Christmas, using exactly the same ending…
But after a bit more research… this appears to be the first - Nat King Cole, The Christmas Song (1946), otherwise known as 'Chestnuts Roasting on an Open Fire'