The Wales centre started outside Owen Farrell in the 24-21 victory and played all 80 minutes of the game in Wellington.
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Despite trailing by nine points in the second half, the Lions were able to fight back and claim the win, and Davies felt they were deserved winners.
He said: “That is my first victory over them in I don’t know how many times trying.
“It was a great result tonight, a good performance in parts and the boys really enjoyed it so we will look forward to next week.
“You can take a lot of confidence from when we played, we were accurate, we shifted the ball and caused them problems, we can take a lot from that.
“It was good to score some tries, to play a bit of rugby even though the conditions were pretty bad.
“It is making sure you don’t get ahead of yourselves, play in the right areas, put pressure and hopefully time will tell.
“The last quarter we grew into the game, we were finishing stronger and gaining confidence from our physicality and confidence on the ball.”
This is how it feels to level up a #NZLvBIL series…#LionsNZ2017 #AllForOne pic.twitter.com/Ft50JbmD98
— British&Irish Lions (@lionsofficial) July 1, 2017
New Zealand lost Sonny Bill Williams to a red card in the first half, but were ahead for most of the game thanks to the boot of Beauden Barrett.
Davies picked out ill-discipline as one of the reasons the Lions struggled initially, but he was delighted to see them finish strongly, particularly against an All Blacks team renowned for their ability to close out matches.
He added: “They were just soft penalties, handing a really good team easy points. It was not the ideal start but the character and confidence we took to grow into the last quarter.
“That is where the All Blacks have been good, taking a stranglehold in the last quarter, we did that this time and it was pleasant to do.
“They have got a great record at Eden Park but with this victory tonight we have momentum, the whole squad is in a good place and we will look forward to an amazing occasion. It is a challenge that we will relish.
“It was great support tonight and hopefully no one is going home and they all have tickets for next week, it is going to be one of those occasions you will remember for your career.”