The Rapids, who finished bottom of the MLS Western Conference in 2015, signed the 27-year-old Albanian forward to a Designated Player contract on Monday.
Gashi scored 32 goals in 57 appearances for Swiss giants Basel over the past 18 months but has now made the move to Denver and the Rapids could not be happier.
"Shkëlzen was our top attacking target this offseason," Rapids technical director and VP of soccer operations Paul Bravo said in team statement.
"He has proven to be a prolific goalscorer in his club's domestic and European successes. In addition he has been instrumental in his national team's qualification for Euro 2016.
"We are delighted to sign him on a permanent transfer and look forward to seeing him in burgundy."
Prior to his stint with Basel, where he scored 22 goal to win the Golden Boot on the way to the 2014/15 Swiss Super League title, Gashi netted 26 goals for Grasshopper between 2012 and 2014 following a prolific spell with Aarau.