Deep Purple has already been mentioned several times, but the one I thought of was their “Speed King”. It begins with a massive multi-instrumental crash followed by a brief bit of what really does sound church organ chords, before the fast hard rock of the main song kicks in.
I realize this is a zombie, but in six years no one has mentioned “Point of Know Return” by Kansas?
Or ABBA? Benny Andersson was something of a child genius on any type of keyboard.
Renegade and Blue Collar Man by Styx. 
Mendocino, She’s About A Mover or just about anything else by Sir Douglas Quintet
One Mint Julep by Ray Charles
Outta Spce by Billy Preston
Whistling Organ, Happy Organ, Catnip, Rinky-Dink all by Dave “Baby” Cortez
Organ music in rock and jazz styles is nearly always generated by electric organs, electromechanical organs, or digital keyboards rather than massive pipe organs. The Hammond B3 electromechanical organ (along with similar models like the B2, C2, and C3) was the most popular choice for keyboardists in the 60s and 70s as these are very versatile in tone with all the tonewheels, drawbars, presets, and variable speed rotating Leslie speaker, which is responsible for the throaty warble sound sometimes heard. The B3 was invented in the 1930s as a smaller less costly alternative to pipe organs in churches and theaters.
The B3 is still somewhat popular today at least among musicians who can afford one but these are certainly used in the studio far more than the stage since modern light weight keyboards like Kurzweils, Rolands, and the Hammond XK series somewhat reasonably mimic B3 sounds. I’ve only seen a full Hammond B3 + Leslie setup at concerts probably five times in the past 10 years. They are portable but require four strong adults and a large van or other roomy vehicle (UHaul truck, 18 wheeler, etc) in order to permit transport. Likewise, I don’t see touring groups using grand pianos very often either.
Hammond B3 organ with Leslie speakers…a brief performance viewed close up
A Leslie speaker cabinet in action
Some of my favorite examples of B3 use in rock music are:
‘Hush’ as covered by Deep Purple
‘Blinded by the Light’ as covered by Manfred Mann’s Earth Band
‘Echoes’ by Pink Floyd (Live at Pompeii version) The extensive instrumental jam starts at 5:09
‘Us and Them’ by Pink Floyd
‘Whipping Post’ by the Allman Brothers Band
‘Dreams’ by the Allman Brothers Band
Real pipe organ music is exceptionally rare in rock music, however.
True, but the OP is calling for Organ music and didn’t specify Hammond, Wurlitzer, or Pipe organ.
I happen to have Santana’s *Evil Ways *playing on my computer right now. Greg Rollie is doing a fine pre-Journey job of some kind of organ. Hammond B3, perhaps? He also did a lot with Journey…People and Places, Walks Like a Lady, etcetera. I rode to work this morning listening to David Bryan doing a fun little intro to Bon Jovi’s Let It Rock.
And I remember an interview in Rolling Stone with Tom Scholtz when Third Stage came out. He was saying the Pipe Organ in [Boston’s Cathedral, perhaps? I can’t remember) was intimidatingly massive and gloriously fun to play. But then, listening to Third Stage, I can only find a single chord at the end of A New World that sounds like a pipe organ.
—G!
There was a momentary scandal when a morning drive
hostess on San Diego radio queued up
Deep Purple’s Perfect Strangers
and closed the show by saying
“That’s it for us, peoples!
Okay, somebody give me an organ!”
Elton John, Funeral for a Friend
Dirty Work from Can’t Buy a Thrill by Steely Dan has a lovely organ line at the start.
The bit near the end of Fake Plastic Trees, the ‘it wears me out’ bit (has a mellotron perhaps) just wanted to mention that, it always gets to me. Beautiful.
Rick Wakeman did use a pipe organ on Yes’ Going For The One album. The song “Parallels” opens with the the sound of Wakeman on the pipe organ.
The OP specifically said
I don’t know if he really intended it to be that restrictive, though–there’s very little rock music involving genuine pipe organ.
I was just listening to Santana’s “Soul Sacrifice” and marveling at the organ, which made me remember this thread. The first Santana album came out essentially simultaneously with Woodstock, so almost everyone there was discovering them. Imagine “Soul Sacrifice” as your introduction to a band. I think it blows away everything else from Woodstock. Props to Greg Rolie and his Hammond B3.
To be honest, I don’t think I realized there were kinds of organs (well, musical ones). 
That’s a great song, but there’s no organ in it. The keyboard instruments are ARP synthesizer (most famously used by The Who on ‘Baba O’Reilly’ and 'Won’t Get Fooled Again) and acoustic piano.
[QUOTE=Exapno Mapcase]
I was just listening to Santana’s “Soul Sacrifice” and marveling at the organ, which made me remember this thread. The first Santana album came out essentially simultaneously with Woodstock, so almost everyone there was discovering them. Imagine “Soul Sacrifice” as your introduction to a band. I think it blows away everything else from Woodstock. Props to Greg Rolie and his Hammond B3.
[/QUOTE]
Props from me as well. Early Santana albums & concerts have awesome Hammond + Leslie performances.
**Might wanna try When the music’s over by The Doors.
Great song.
**
Wolfmother totally nail that early 70s Deep Purple/Led Zeppelin/Black Sabbath sound, and they do love them some hard rock organ: here’s Mind’s Eye, with epic organ solo at 3.16.
I Can’t Let You Go sung by Rainbow has an organ solo opening.