Williamson said he knew the team put a lot of importance on test cricket so win the championship was "very exciting".
"It's a very special thing. I know the boys put a lot of kudos on test cricket, absolutely ... so it's very exciting.
India started the day at 64-2, leading by 32 runs.
Victory required a stunning bowling performance and that's what they delivered, with India all out for 170.
The Black Caps then had to chase down 139 runs and got there - with Ross Taylor hitting the winning runs - with 7.1 overs to spare.
Asked whether he stayed up and watched the whole game, Williamson senior said they ended up going to bed as they weren't sure whether they wanted to watch it; especially with memories of Lords 2019 in the back of his and wife Sandra's minds.
"Last night we went to bed not sure whether we wanted to watch it as three results were still on the cards and memories of Lords of 2019 started to flood back, so we were just hoping and praying that something positive would come out of this.
"It was pretty sleepless. In the end I checked my phone and I saw all the texts that said 'you better get up and watch this', and that was about 4.30am.
"It's very exciting and we're ecstatic for Kane and the team, it's absolutely fantastic."
