The 24-year-old Denmark defender becomes Leeds' second signing since their Premier League status was confirmed for another season after a dramatic last-day win at Brentford.
He joins former team-mate and United States midfielder Brenden Aaronson in switching to Elland Road from RB Salzburg in a deal believed to be in the region of £10million.
"When I heard about the interest I looked at the project and what was going on and I was never in doubt, I was 100 per cent convinced," Kristensen told the official club website.
"It's a huge historical club which is playing in the best league in the world and for me, it fits everything that I love about a football club."
He said: "I see myself as a physical, emotional power player, I like to go forward, run up and down the flank and I have a big engine for that.
"I'm a guy who puts 100 per cent effort in every time I'm on the pitch and I'll die with my boots on."
Marsch, who coached Salzburg from 2019-21 before moving to their sister club RB Leipzig, signed Kristensen from Ajax for Die Roten Bullen in 2019.
The American said he has total faith in Kristensen making a successful switch to the Premier League and is convinced he will hit the ground running.
Marsch said: "Rasmus is a player who developed a lot over the last three years.
"He established himself at Red Bull Salzburg as one of their best players, a very good Champions League full-back, and a Danish international.
"I am totally convinced that he will continue to grow into a great Premier League full-back as well."