Yet another one: “Baby, It’s Cold Outside” first appeared in the 1949 film Neptune’s Daughter.
It was used in the documentary The War at Home, which was released in 1979 or so.
It wouldn’t surprise me if people thought United Airlines commissioned Rhapsody in Blue rather that it be a 1924 George Gershwin composition.
I think of hippos and crocodiles.
Its soundtrack contained two other songs which have also endured: “Happy Holiday” (which, like “White Christmas,” has endured in the Christmas music songbook) and “Easter Parade.”
“Hooray for Hollywood” first appeared in the 1937 film Hollywood Hotel.
Powerhouse premiered as an instrumental composition on CBS Radio’s Saturday Night Swing Club in January or early February 1937. Now it’s famous for its use in Warner Brothers and other cartoons.
The Dating Game!!
The jazz composer Raymond Scott wrote a short, rather odd, instrumental that would have been long since forgotten, except that parts of it were used in Warner Brothers cartoons:
Raymond Scott - Powerhouse [mp3]
If you are not the patient type, skip to 1:15
ETA: Ninja’d by nine minutes
Jean-Joseph Mouret’s Rondeau is "The Music from Masterpiece Theater
In 1969 a minor soul band called the Winstons recorded an instrumental track called “Amen Brother”. It was too short, so they added a 7-second drum break to the middle. You can hear it here at 1:26.
If it sounds familiar to you, that’s because for some reason, hip-hop artists started sampling it in the mid-1980s. You probably recognize it from “Straight Outta Compton”, but it was sampled over and over again. Thousands of time. It is literally one of the building blocks of hip-hop.
The Warsaw Concerto is a popular classical piece. It was written for the movie Dangerous Moonlight, which is completely obscure.
I actually only realized that very recently. 
I heard that music in some show or commercial and commented “Oh that’s the United theme!” only to have my mother be like “…no… that’s Rhapsody in Blue by Gershwin!”
I always fly United and have heard that so many times! I honestly had no idea. 
Many old children’s songs/lullabies
I think I’d Like To Teach the World to Sing outlived the Coke commercial by a few months.
“As Time Goes By” was written in 1931 for a Broadway show.
“Georges Bizet’s “Les Toreadors” from Carmen Suite No. 1 becomes an iconic theme for a podium celebration in Motorsports, especially Formula 1.”
The Champagne Spraying Song
Elton John’s song “Philadelphia Freedom” long outlasted the professional tennis team of the same name that it was written in honor of.
And, for that matter, it has outlasted the popularity of tennis as one of the most popular spectator and participation sports in the U.S. Tennis is not gone, of course, but it’s much more of a niche (both for play, and for watching on TV) than it was when that song was released in the '70s.
The song “Singin’ in the Rain” has long outlived its origin in The Hollywood Music Box Revue."
Yeah, I don’t think I realized until I saw it that Singin’ in the Rain, the movie, is almost entirely pre-existing songs.
Make 'Em Laugh and Moses Supposes are the only new songs in the movie.