Here's the list I compiled ages ago, which has a couple of uncertainties:
Alexander Courage:
"The Cage"
"Where No Man Has Gone Before"
"The Man Trap"
"The Naked Time"
Courage also did the "log entry" music in ST:TMP and worked as Jerry Goldsmith's orchestrator on FC and INS.
Credited to Courage, actually ghostwritten by Scott Huston:
"The Enterprise Incident"
"Plato's Stepchildren"
(There's no verified proof of this, but Huston was a friend of my father and mentioned this to him and to the local newspaper; also they're in a distinct style from Courage's work in the pilots and first season.)
Fred Steiner:
"The Corbomite Maneuver" (partial score)
"Mudd's Women"
"Charlie X"
"Balance of Terror"
"What Are Little Girls Made Of?"
"The City on the Edge of Forever" (partial score)
"Who Mourns for Adonais?"
"Mirror, Mirror"
"By Any Other Name"
"Elaan of Troyius"
"Spock's Brain"
And possibly:
"The Omega Glory" ("Star-Spangled Banner" arrangement)?
"The Way to Eden" (songs, music only)?
(These episodes have no other original music.)
Steiner also scored TNG: "Code of Honor," the only TOS composer to cross over.
George Duning:
"Metamorphosis"
"Return to Tomorrow"
"Patterns of Force"
"And the Children Shall Lead"
"Is There in Truth No Beauty?"
"The Empath"
Gerald Fried:
"Shore Leave"
"Catspaw"
"Friday's Child"
"Amok Time"
"The Paradise Syndrome"
Sol Kaplan:
"The Enemy Within"
"The Doomsday Machine"
Jerry Fielding
"The Trouble With Tribbles"
"Spectre of the Gun"
Joseph Mullendore
"The Conscience of the King"
Samuel Matlovsky
"I, Mudd"
Desilu music supervisor Wilbur Hatch wrote the music to the song "Beyond Antares" in "Conscience of the King." Ivan Ditmars composed the Brahms paraphrase in "Requiem for Methuselah." The harpsichord music in "The Squire of Gothos" includes the Sonata in C Major, K.159 by Domenico Scarlatti and Roses from the South by Johann Strauss II.