Gonzalo Higuain's injury-time strike was all that separated the two teams at the Stadio San Paolo in mid-week cup action.
With the match looking headed for extra time, Higuain broke Inter's stubborn resistance by shaking off captain Andrea Ranocchia and firing the ball beyond Juan Carrizo.
Mancini was bitterly disappointed by the manner in which Inter conceded, having seen five of his defenders fail to deal with the Argentina international.
"I don't have much to say, you just can't concede a goal like that," said Mancini.
"We played really well throughout the game but every time we go and hand the opposition a free goal.
"We were far too lax. Higuain is a smart player and he nipped in there before anyone else on the throw-in.
"We were like a bunch of chickens, it was five against one. It was a ridiculous goal to concede."
Mancini added: "We're growing as a team but we can't afford to make so many mistakes.
"We did the same on Sunday with [Simone] Zaza and [Nicola] Sansone. This goal we let in tonight was absurd. I'm shocked. We shouldn't have lost the game like that."
Inter goalkeeper Carrizo urged his team-mates to keep working hard in the midst of a three-match losing streak in all competitions.
"It's hard to find the right words. We can't fathom why the results just aren't coming for us," Carrizo told Inter Channel.
"It was a good performance and it's hard to work out the reasons behind this spell we're going through.
"I should be happy with my own display but I can't feel happy tonight. We wanted to reach the semi-finals and we couldn't manage it.
"We have to take the criticism on the chin, get our heads down and knuckle down."
