The 22-year-old went to the USA on trial after his Tynecastle contract expired at the end of last season and has signed a two-and-a-half-year contract.
Hearts confirmed they were not entitled to compensation under Fifa rules but a sell-on clause has been negotiated into the deal, believed to be at Nicholson’s request.
The winger scored 16 goals in 127 appearances for Hearts.
The former Scotland Under-21 international said: “It was an easy decision after speaking with the coaching staff. Everyone made me feel comfortable and at home straight away.
“It’s a great club and I am excited for the new challenge ahead in MLS and I cannot wait to play in front of all the amazing fans.”
Nicholson has been signed by former Stoke and Everton forward Adrian Heath, who is the Minnesota head coach.
Heath said: “We are really pleased to have added a player with the skill set of Sam. He’s a player who can cut inside from either side of the pitch. He’s got two really good feet and he’s got some goals in him. He can beat his man.
“It’s not very often that you get the opportunity to bring in a player from Europe who looks at MLS as a step up. We are delighted to have him and we think he will be a great addition to our squad.”
l Partick Thistle goalkeeper Tomas Cerny has signed a one-year contract extension. The 32-year-old Czech is now tied to the Firhill club until the summer of 2019.