The Red Devils dominated in a goalless first half where Marouane Fellaini and Marcos Rojo hit the woodwork.
But they were shell-shocked at the death as Michael Lang turned home Raoul Petretta cross to spark wild celebrations from the home faithful.
But Mourinho was less than impressed with his side’s inability to guarantee their place in the knockout stages.
United top Group A, but Basel and CSKA Moscow are three points behind, meaning qualification will go down to the final game.
The Red Devils must avoid a 6-0 defat against CSKA Moscow to guarantee a place in the last 16.
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“We lost because in the first half we should be 5-0 up but we could not score one goal after being so dominant,” Mourinho told BT Sport.
“It looks for many periods in the first half it was impossible not to win.
“I think the majority of the second half the feeling of everyone was OK, it will be 0-0, but they got the late goal.
“It is not fair for (Daley) Blind to blame himself. He had a very positive game. We have to blame everyone on the pitch because we had so many chances.
“We don’t need to win the (final group) match, we don’t even need to draw, but obviously we will try to win.”
United return to domestic action on Saturday with a home clash against Brighton (3pm).
They will be hoping to close the eight-point gap on league-leaders Manchester City, who travel to Huddersfield on Sunday (4pm).
But Roy Keane insists United can win the Champions League despite faltering against Basel last night.
“The big plus for United is that their form has been that good in the Champions League that they gave themselves room to slip up tonight,” Keane told ITV Sport.
“When United have their strongest team out, they will compete with a lot of the teams in Europe.
“There are five or six players that played tonight that won’t be in their strongest team.
“Mourinho is looking at the bigger picture – forget about tonight and move on.
“They’ll be competing to win this title without a doubt.”