They’re Playing Our Song is the title of a musical (book by Neil Simon, lyrics by Carole Bayer Sager, and music by Marvin Hamlisch) that opened on Broadway on February 11, 1979. However, “they’re playing our song” was a clichéd saying long before this. “They’re playing our song” was identified as a Hollywood movie cliché by at least 1947; a couple would be seen at a table, one character would say that the band was playing “their song,” and they’d get up to dance.
 
“Listen, there’s the Morrow hotel orchestra. They’re playing our song” was cited in print in 1931.
 
   
Wikipedia: They’re Playing Our Song
They’re Playing Our Song is a musical with a book by Neil Simon, lyrics by Carole Bayer Sager, and music by Marvin Hamlisch.
 
In a story based on the real-life relationship of Hamlisch and Sager, a wisecracking composer finds a new, offbeat lyricist, but initially the match is not one made in heaven. The two undergo a series of trials and overcome a number of hurdles before finding true love by the final curtain.
 
They’re Playing Our Song is essentially a two-character show. Vernon and Sonia are the sole characters on stage; each character has a three-person Greek chorus acting as their inner voices, and there are no big production numbers.
 
5 July 1931, Salt Lake Tribune (Salt Lake City, UT), “Mary Faith” by Beatrice Burton, pg. 8D, col. 7:
“Listen, there’s the Morrow hotel orchestra. They’re playing our song.”
 
17 September 1947, Canton (OH) Repository, “New Movies Use Same Old Dialog” by Aline Mosby (United Press), pg. 7, col. 2:
“Hollywood,” he painfully admitted, “produced 425 feature pictures last year.” In 419 there were these lines:
(...)
Love scenes, he discovered with horror, are just variations on an old theme. The characters always get around to:
 
“They’re playing our song.”
 
22 February 1948, Coshocton (OH) Tribune,  “In Hollywood” by Erskine Johnson, pg. 11, col. 1:
Some of the clichés, as Englund sees them:
 
ROMANTIC DIALOG AND LOVE STUFF: “Listen, darling, they’re playing OUR song.”
 
OCLC WorldCat record
They’re playing our song
Author: Billy Butterfield
Publisher: Camden, N.J. : RCA Victor, s1957.
Edition/Format: Music LP : English
 
Google News Archive
10 February 1958, The Free Lance-Star (Fredericksburg, VA), “Senator Soaper Says,” pg. 6, col. 3:
Scientist and his wife listening to the “eeee” of the radio signal from the Explorer: “Dear, they’re playing our song.”
 
OCLC WorldCat record
It’s time to dance : they’re playing our song.
Author: Fred Ellis; Band of Yesterday.
Publisher: Los Angeles, Calif. : Barbary Coast Records, [1959]
Edition/Format: Music LP : Popular music : No Linguistic Content
 
Google News Archive
6 April 1961,  Milwaukee (WI) Sentinel, Earl Wilson syndicated entertainment column, pt. 1, pg. 19, col. 1:
DELIGHTFUL gag around is that when the Joseph Kennedys stepped into the Golden Pheasant, the cash register rang, and Mrs. K. said, “They’re playing our song.”
 
OCLC WorldCat record
They’re playing our song
Author: Jan Garber
Publisher: New York : Decca, [1964]
Edition/Format:Music LP : Popular music : No Linguistic Content
 
OCLC WorldCat record
They’re playing our song
Author: Al Hirt; Marty Manning
Publisher:[New York?] : RCA Victor, [1966]
Edition/Format: Music LP : Popular music : No Linguistic Content
 
Google Books
They’d Hang You in Nashville
By William Gleason  
Chicago, IL: Dramatic Publishing Company
1976
Pp. 24-25:
(Spanish dance music starts. )
STAN (reaching out to an imaginary lady). Mrs. Worthington? They’re playing our song!
 
Google Books
The Franchiser:
A Novel
By Stanley Elkin
New York, NY: Farrar, Straus & Giroux, Inc.
1976
Pg. 183:
What, you never heard the expression ‘They’re playing our song’?