Well here is the source on Genjutsu:
Take a deep breath....
This confirms that Genjutsu is the manipulation of the chakra flow (which is connected to the nervous system) in the brain. This controls sensory information only.
The Genjutsu user "traps" their opponent by sending out their own chakra to manipulate the flow in the opponent's brain (think puppet jutsus but a lot more subtle).
Getting out of a Genjutsu requires you or someone else to further manipulate your internal chakra flow. This would cause the flow to become less predictable and the original manipulator would be unable to control it, thus wasting chakra in the attempt.
The Kyuubi is probably taking this all up a notch (Similar to Ino's technique). It completely overwhelms Naruto's regular chakra flow to the point that it takes over. This is also probably why it harms Naruto, as too much chakra is passing through him.
SPIRIT + BODY = USABLE CHAKRA.
Both spirit and body must be present to function. No spirit is a coma. No body is death. Genjutsu interferes with a specific link between the body (the sensory part of the brain) and spirit.
Naruto's spirit is pushed aside and replaced by the Kyuubi's thus Naruto isn't comatose. And the body acts with the the new spirit. Naruto probably isn't cut off completely most likely due to the seal. Therefore he can come back. Also the kyuubi is connected with the seal, thus it can connect in such a fashion.
I can only think of two reasons why the Kyuubi's control doesn't last.
1) The seal - This is, on some level, under Naruto's control, like some sort of mental on/off switch. At any point Naruto can turn it off (sub-consciously only it seems right now) or there is a sort of time limit.
2) Physical Strain - With chakra manipulation there are 2 components (3 with sage mode) that must remain in some sort of balance in order to function. It doesn't have to be perfect to make it work, but there are side effects when unbalanced (becoming a frog with too much nature for example).
The body has only so much stamina. If you use too much of the spirit or if you don't have enough stamina, pain or eventually death occurs. Because the fox needs Naruto's body to stick around it can't kill him, so it has to go away so Naruto can recuperate.
On my next broadcast (heh) The Grand Chakra Theory with graphs and junk!
Bleh, such a mouthful and I still have more to say that is on topic.
I really see two potential outcomes of sage training relating to interacting with the Kyuubi.
1) Kyuubi replaces Naruto's spirit component, thus allowing him to compensate for more Nature chakra. This would in turn allow Naruto to draw out more of the Kyuubi's power before his body shuts down.
The downside to this method is that I don't believe Naruto is in control at this point and the Kyuubi is no sage. (For that matter neither was Minato as far as I know, so I don't think this is "That Jutsu").
2) Kyuubi replaces the nature chakra, keeping Naruto in control. It may be easier to achieve (as it seems Naruto has done it before untrained) and easier to sustain.
The downside may be that replacing nature chakra may not be possible, although I think it likely.
So perhaps sage training is a primer on Kyuubi management?