We may have different views about the place of horns in rock music. My rule is no tootling, and I’m unalterably opposed to saxophones. Except in “Baker Street.” And Morphine. And the Climax Blues Band. And James Brown. There may be other exceptions.
Anyway, if the saxophone has a proper role, it’s a strictly limited one as regards the number of notes played.
The greatest example of sax discipline I know is “I’m Shakin’” by Little Willie John, memorably covered by The Blasters in 1981 (which is where I started with it). There’s no yakety sax distraction here, just a super-simple, maximally coercive riff. The song couldn’t be stripped down any finer: it’s been filtered and strained and blow-torched until all that’s left is the bones and tough sinews of an indestructible body. (Little Willie John did the original “Fever,” too, which has a similar quality. He’s still “got fever” in this one.)
The riff can be done acceptably by electric guitar (check the Jack White version of 2012), but the sax nicely contributes some cultural age, sending a taproot into the 50s or 40s. It seems like our parents and grandparents would have heard it. But there’s nothing in it to make it dated. It ought to wheel back around every few decades.
Postscript #1. “Wooly Bully” (Sam the Sham and the Pharaohs, 1965) has an OK sax part, making all due allowances, but its claim to sax fame for me lies in its transition to the sax break at 1:20 with an outstanding scream by Sam the Sham that mimics a sax. I’ve never heard a scream so like a sax.
Postscript #2. This song has an utterly different attitude and idea than “I’m Shakin’” but a very similarly appealing simplicity and insistence: Michelle Shocked, “Quality Of Mercy” (from the Dead Man Walking soundtrack, 1996).