Paulo Dybala’s 18th-minute goal settled the contest, with Juventus enjoying the majority of possession and chances but United improving in the second half, when Paul Pogba hit the post.
The result leaves United five points adrift of Juve in Group H ahead of their visit to Turin next month and two points ahead of Valencia, who Mourinho described as his team’s real rivals.
The Portuguese said: “We play against one of the biggest, really biggest candidates, to win the Champions League and we did the possible.
“We could get the different result, especially because of the way we played in the second half. We didn’t especially because Mr (Leonardo) Bonucci, Mr (Giorgio) Chiellini, they could go to Harvard University to give classes about how to be a central defender. Absolutely fantastic.
“It is a team that has everything, they have talent everywhere. Of course Cristiano (Ronaldo), Dybala, the talent they have in attack, but the quality all over the pitch and a really difficult match.
“I’m happy with the players. Since the draw we knew it would be a fight with Valencia for the second position and that is what we are going to try.”
Mourinho was frustrated by United’s summer transfer dealings and cited Juve’s ambition in the market as the reason they are among the favourites for the biggest titles.
Asked how far away United are from the Italians, Mourinho said: “A bit far. Juventus are champions (of Italy) for seven years in a row, two Champions League finals in the last four or five years and (they are) not happy with what they have, they want more.
“They had (Gonazlo) Higuain, (Mario) Mandzukic, Dybala, they want more. They want Ronaldo. They had (Andrea) Barzagli, Chiellini, (Daniele) Rugani, they are not happy, they want more, they want Bonucci, and they go for the best players in the world.
“How can you reach this level? It’s not easy. We work with what we have. We try to improve.”
Mourinho praised the performance of his defence, especially Victor Lindelof, who put in his best display in a United shirt, but admitted he is concerned about the form of striker Romelu Lukaku, who has not scored for United in eight games.
The United boss said: “No criticism at all, a complete understanding the player is a fantastic pro that wants to give everything but I have to agree his moment is not sweet, not just with the goals he is not scoring but in his confidence, movement, touch, he is not linking the game well with the team – but he’s our striker and he’s a good striker and a striker we believe in.
“He is a hard-working guy, one day the goal will arrive and the confidence will be back.”