"A few things to be thankful for. Firstly, the circumference and strength of my neck, I may not be standing without it," Cane said.
"Secondly, the world class medical and surgical team I've had here in Pretoria, couldn't have asked for anything more."
The All Blacks' official Twitter account earlier posted an update after Cane's operation.
"@SamCane7 is up and about after his successful operation in South Africa. Sam would like to thank everyone for all the support he has received from around the world, he is set to make a full recovery," the Tweet said.
After a collision with Springboks loose forward Francois Louw in the 35th minute of the All Blacks comeback victory, Cane stayed down and then looked in obvious pain.
He needed surgery to repair a fracture on the lower right side of his neck and will remain in South Africa for a week to recover.
All Blacks doctor Tony Paige said there was no nerve damage but he will stay to monitor Cane's progress.
[All Blacks aftermath: The Houdini on the Highveld - All you need to know about ABs dramatic comeback]
While expected to recover, Cane will be out for at least three months and, therefore, misses the northern tour which opens with the third Bledisloe in Yokohama in three weeks.
"First he's got to get through the operation," Paige told media in South Africa. "We have every confidence he'll do well with that and then like any fracture it takes a couple of weeks and months. It will probably take about three months for the bone to get strong but we'll just take it step by step.
"The operation is really just to keep everything in the right place so the bone can heal on its own accord."
Like All Blacks captain Kieran Read following his spinal surgery late last year, Cane's return to play will be a cautious affair.
"Many sportsmen have had this injury and people in other walks of life and they usually make a good recovery. It's one level of the neck.
"Sam's in good spirits he's obviously sore in the middle of the night as you are the night after a fracture but he's being well looked after and is getting lots of support and texts and phone calls."
Given concern around a break in the neck, All Blacks coach Steve Hansen attempted to downplay fears.
"Once you remove the drama of it being a neck injury you've just got to look at it and treat it like any fracture," Hansen said.
"It's obviously really disappointing for Sam and his family but the pleasing news is he's being given great medical care from the time of the injury through the whole process and best of all he's going to get a full recovery and play again."
Cane's injury weakens the All Blacks loose forward stocks but the impressive form of Ardie Savea should see him step straight into the No 7 jersey.
With Matt Todd playing under Robbie Deans' Panasonic Wild Knights in Japan, Blues and Auckland prospect Blake Gibson probably heads the list of contenders to replace Cane on the northern tour.
Highlanders opensides Dillon Hunt, a mid-week tourist last year, and James Lentjes, who just made his return for Otago, are other options.
"We'll have to get another one. Who that is yet we're working through the process. We've got to get one to replace Sam whether we take three on the tour that is another question."
