Lazio came from behind to defeat Udinese thanks to goals from Alessandro Matri and Danilo Cataldi, setting up a tie with Juventus, the side who beat them in the final in May.
Like city rivals Roma, Lazio headed into their tie on the back of a long winless run having not picked up a domestic victory since October.
However, while Roma were humbled by Serie B outfitSpezia on Wednesday, Lazio secured a dramatic triumph.
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They fell behind in the 67th minute when Panagiotis Kone volleyed home, but equalised in controversial circumstances in the 70th minute when they played on despite an injury to Udinese’s Rodrigo Aguirre and Matri fired past impressive teenage stopper Alex Meret.
Having used all three substitutes, Udinese played the final 20 minutes with 10 men and fell behind when Cataldi headed past Meret in the 79th minute.
“I’m angry with (Lucas) Biglia and the other Lazio players,” Udinese coach Stefano Colantuono told the Udinese TV Channel.
“For me, they didn’t act correctly. We thought that having put the ball out for Aguirre’s injury, they’d kick it back to us, but they didn’t.
"I think they could have won without these tricks.”
Milan sealed a last eight meeting with Carpi after goals from M’Baye Niang and Carlos Bacca saw off Sampdoria and extended their undefeated streak to seven games.
Niang opened the scoring in the 50th minute when he was played in by Bacca, attacking at searing pace, and finished assuredly past Emiliano Viviano.
Sampdoria, who have now won just one of their last 12, were reduced to 10 men in the 65th minute when Ervin Zukanovic received two quick-fire yellow cards, and Bacca added a second in the fourth minute of stoppage time after evading Niklas Moisander and finishing from close range.
“We simply had to win tonight and we achieved what we came here to do,” Milan coach Sinisa Mihajlovic said to the club's website (www.acmilan.com).
“The final is certainly within our reach and tonight’s win is a confidence boost for our league campaign.
“We’re all aware that Milan can and must do better.”
(Reporting by Ed Dove; editing by Justin Palmer/Toby Davis)
