The Whitley Bay striker, 22, had trained with City for the first time just 24 hours before Tuesday's Central League North game against Hartlepool United at Bootham Crescent.
But his performance in the 5-2 victory prompted the club to snap up his services immediately on a deal until the end of the season.
"I am absolutely buzzing and it's all been a bit of a whirlwind," he told the club`s press officer Ian Appleyard. "I spoke to the Gaffer straight after the game and he has put his faith in me."
Kempster has played over 200 games for Ebac Northern League side Whitley Bay but has never previously played for a full-time club.
He will now be giving up his jobs as a PE teacher and barman to become a full-time professional with the Minstermen.
"It's a massive decision but you have got to try and get your foot on the ladder," he said. "If someone is giving you a professional contract, you have got to take it. It's what every lad dreams of really."
Kempster scored City's equalising goal in the 52nd minute with a bullet header into the roof of the Hartlepool net.
He added his second 8 minutes later when getting clear of the visitors' defence and stroking the ball past the goalkeeper.
He completed his career-changing hat-trick in the 59th minute when curling a low shot around the goalkeeper from just inside the penalty box.
By the end of the afternoon, a section of City fans on social media were clamouring for the club to sign him up!
"It's a really good set-up, a really good ground, and the fanbase is unbelievable," he said. "I can't wait to get out there and score some goals".