Cane went for an HIA at halftime in Yokohama, and passed it, but he did not return to the field with his teammates for the second half.
Match officials determined Cane had spent too much time off the field and so could not return, despite passing the concussion test.
According to World Rugby regulations, if a player is unable to return from an HIA after 10 minutes, or directly following the 15 minute halftime break, then he must be replaced.
The All Blacks say Cane could not get back from his HIA assessment in time because of the distance he had to walk within Yokohama Stadium for the testing.
Although they stress the protocols were followed correctly on Saturday, World Rugby has now modified the rules to only start timing the HIA once the player has entered the testing room.
"Specifically for Rugby World Cup, with immediate effect, the official 10-minute HIA window will commence from the moment the temporarily replaced player enters the HIA room, rather than when the player leaves the playing area," World Rugby said in a statement on Sunday.
"This is to facilitate slightly longer distances to the HIA room from the field of play at some of the venues."
"World Rugby has also reiterated that if the time allowed for a temporary replacement elapses during halftime, the replacement shall become permanent unless the replaced player returns to the field of play immediately at the start of the second half."
Hansen welcomed the change.
"It wasn't through any fault of his that the test took longer than it should have ... it only took as long as it needed to take but it wasn't anything we could control.
"So, I think they have worked out - it isn't common sense is it? We are here to look after the athlete and so they have modified.
Cane, who cut a frustrated figure on the sidelines after being denied the chance to return to the field, was replaced by lock Patrick Tuipulotu, forcing Scott Barrett to move to the back row alongside captain Kieran Read and Ardie Savea.
"I am just glad they have sorted it out. I wasn't overly happy about it last night when one of your best players is missing 40 minutes of the game," Hansen said.
"But it is what it is; they have shown good leadership, they have acknowledged that they haven't got that part of it right and they have changed it. Can't ask for anything more."
Fortunately, Hansen could call upon Savea, who dominated the breakdown despite playing in all three positions across the backrow at some point during the match.
"He is playing really well isn't he?" said a beaming Hansen.
"He has just got better and better as he has got older and stronger.
"We have always known he is an exceptional athlete but at the moment he is at the peak of his powers.
"It gives us a good bit of X factor there."