The 24-year-old Australia international has put pen to paper on a three-year deal and has the option of a further 12 months.
“I couldn’t be happier and I’m just excited to be starting the next chapter of my career,” said Irvine, who scored in Burton’s 4-1 defeat at the KCOM Stadium earlier this month before being sent off.
“It has all happened relatively quickly, but once I found out about the interest from Hull City it was an easy decision even though leaving Burton was difficult in its own way. I just can’t wait to get started now.”
Jackson, born in Melbourne, joined Celtic’s academy in 2010 and made one senior appearance for the club before loan spells at Kilmarnock and Ross County.
He signed a permanent deal with Ross County in 2015 and Burton paid an undisclosed club record transfer fee to take him to the Pirelli Stadium in July last year.
In his one season with the Brewers, Irvine scored 10 goals in 44 appearances.
Hull paid an undisclosed fee for Wolves striker Nouha Dicko on Tuesday and Irvine is their 11th new signing since they were relegated from the Premier League last season.