Despite dominating the majority of the match at St. Jakob-Park, Michael Lang scored a late winner to keep Basel's last 16 qualification hopes alive.
Cottee believes United's dip in form during the second half can be partly blamed by the 66th minute substitution of Paul Pogba.
Talking to 'Sky Sports', he said: "It was possibly a mistake to take Pogba off, he was probably their best player.
"Mourinho made changes and brought on Marcus Rashford, Nemanja Matic and Zlatan Ibrahimovic, but Pogba was the one in midfield who was making things happen.
"They did lose the midfield in the last 20 minutes, but United should have had the game wrapped up."
Did Mourinho take Pogba off to early? Let us know what you think in the comments.