The 32-year-old, who previously coached Chelsea's Under-23 side and was an assistant coach with the Wales national team, will leave United after their Premier League game against rivals Manchester City on Sunday.
"I'm incredibly excited to be joining a club with such a strong football culture, a fanatic fan base and a brilliant infrastructure," Ramsay said.
"I've spoken to a lot of the club's staff throughout the process and you can't help but feel everyone's passion for moving the club forward."
Minnesota won their opening league game and sit third in the Western Conference, level on points with leaders Portland Timbers.