Fredy Guarin's goal was enough to see Inter progress.
"There was a big risk at the start, as a goal then would change the game," Mancini told Mediaset Premium.
"With the red card, it became easier. We are getting better, but still make mistakes in every single match. When we don't make mistakes any more, then we can say we're better off.
"We should've scored earlier to relax into the game. Guarin has always been able to shoot like that. Maybe in the past he got too bogged down carrying the ball, but his runs from deep are dangerous.
"We must improve in every respect. We are making progress and will try to run fewer risks in defence, ensuring the midfield protects them more.
"Celtic were more locked down today and didn't allow us the same spaces we had in Glasgow. When they attacked, we weren't quick enough to cover the gap between midfield and defence, as Hernanes and Guarin stayed too far forward."
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