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Wagner was hopeful of playing in the first test in Wellington, but as the wait went on in Tauranga hospital, he ruled himself out of contention, before Olivia was born on Wednesday.
The 33-year-old was delighted to be able to be there to welcome his daughter into the world.
"If she was going to be early then I definitely would have been there with the team, but so be it, she wanted to try and stay a little bit longer and give her dad a little bit of a hard time," Wagner laughed.
"We tried everything we could but she didn't want to budge. Without a doubt on her 21st and her wedding day I'll remind her about it."
Wagner, who will come back into the Black Caps XI for the second test at Hagley Oval this Saturday, was also thrilled to be able to support wife Lana, and give back for all the sacrifices she had made throughout his career.
"I wouldn't change it for the world – it was amazing to be there for the birth of my first child, to be there with my wife and support her all the way. She's been phenomenal throughout my career, so to be able to be there for her as well and support her with this has been pretty special.
"I'm very happy – amazing to be there and something I'll never regret."