Beauden Barrett's brilliant solo try sealed the silverware for the visitors at Johannesburg's Ellis Park, as the All Blacks could not be caught after earning a bonus point.
Hansen was relieved with the win but extremely proud of his players who come away from Ellis Park as victors for the first time in almost 18 years.
"There are a lot of reasons why we shouldn't win this game," Hansen said.
"Firstly the travel and the short week and everything but this is probably the best one I guess because the game was so special.
"Both teams were really playing well and when you get a game like that, the team that ends up winning it deserved to win and I think we deserved to win today and played well enough to win.
"It's like any moment at the end of the game when you've won, when you've known you've had to work really hard for it."
All Blacks skipper Richie McCaw said the free-flowing game led to a few tired legs, although he said playing in such a decisive Test match was what he and his team-mates live for.
"We had to dig pretty deep, I think both teams really wanted to play and because of that there was a hell of a lot of running and open play for long periods," McCaw said.
"But it is days like today, that are the reason you put on the boots.
"We all knew leading up to this match that this is one of the ones you want to play and it is why you play the game.
"I was just pretty thankful to have a chance to be out there.
"I think we took our opportunities and when they were dished up we took them.
"There were times we were under a lot of pressure but we got a critical turnover and capitalised on that so it was probably the difference really."