Lake's side edged the three-time Rugby World Cup winners for a first victory over their opposition in 34 attempts.
Wales have suffered a dire run of results since the 2023 World Cup including more than two years without a home Test win, with a host of off-field issues overshadowing the sport.
Last Saturday they sluggishly edged Japan to start their November leg of the Nations Championship on a positive note.
"I think the bad times drive you to want more and also it taught us we can be in a game in so many different ways," hooker Lake told Welsh television channel S4C.
"Last week's scrutiny was well-deserved.
"We had a few difficult chats in the week, to put things to right.
"I'm very proud how the boys answered the questions from last week," the 27-year-old added.
Lake was asked how the squad would celebrate and he jokingly responded: "An early night".
Wales' win was orchestrated by head coach Steve Tandy, who was appointed midway through last year.
"It's not about me but the players and the supporters," the former Ospreys flanker said.
"The effort the boys have put in over the last 15 months or so, they got what they deserved which was an unbelievable performance," he added.
Tandy's side started both halves clinically, scoring twice within the first seven minutes of the first interval and two times nine minutes after the break.
"They had the intent to go after the game. When New Zealand came back after half-time it would have been easy to doubt themselves," Tandy said.
"What I loved about it was at half-time they wanted to chase down the game again.
"The last few minutes they looked really composed.
The boys have created history, they have to celebrate and enjoy the moment," he added.