The Chile international, 29, set up two goals on his debut for his new team in a 4-0 FA Cup fourth-round win over Yeovil.
“He is a fantastic addition,” Mourinho told BBC Sport.
“His choice for the second goal is a choice that a player like Marcus Rashford is too nervous to make.”
He added: “Alexis brings maturity and class.”
Sanchez signed for United on Monday with Henrikh Mkhitaryan heading in the opposite direction to Arsenal.
With Zlatan Ibrahimovic acting as back-up to £75m summer signing Romelu Lukaku in the central role, Sanchez is expected to compete with the likes of 20-year-old Rashford, Anthony Martial, Jesse Lingard and Juan Mata for a role just behind.
Sanchez had a hand in his team’s first two goals in the win at Huish Park, bundling the ball towards Rashford for the opener before slipping a well-weighted pass into the path of Ander Herrera for the Spaniard to drill home.
Mourinho insisted that his squad would be galvanised by the increased competition.
“Good players want good players,” he added.
“The question is always which one is going to be left out, it is not about that for us.”