The Swiss went solo to the finish line at the Snowbird Ski Area and Summer Resort to claim the honours on the Queen stage - and usurp his Garmin-Sharp rival from the top of the general classification in the process.
Tschopp caught a small breakaway on the final climb of the 165km slog from Park City to win before easing ahead to win the stage by 43 seconds from NetApp's Leopold Koenig.
Bontrager-Livestrong's Joe Dombrowski was third, 47 seconds off Tschopp's pace, while Vande Velde finished 1:38 back. The American ended the day in second on the GC, 38 seconds behind the new leader.
The race concludes on Sunday with the King stage, a 123.5km mountainous ride that ends in Park City.