Milan, the nominal home side at a boisterous San Siro, had the better of the play and opportunities but could not convert their chances into goals and head into next month’s second leg level with their local rivals.
The away goals rule is still in force in the Italian Cup, meaning Milan will have a small advantage when they are hosted by Inter for the decisive clash.
“We played well, it was difficult but we have another leg to play,” said Theo Hernandez.
“We’re giving our all and having a great season. We need to keep going and win. We had a lot of chances, but if you don’t score you’re not going to win.”
Serie A champions Inter were poor and have now failed to win in their last five matches in all competitions in a dip in form which has also cost them their lead in the league.
Inter, who are also goalless in four games, are battling for the Serie A title with Milan and Napoli, who are both two points ahead of Simone Inzaghi’s side and face off in Naples on Sunday.
“We’re still the best attack in Serie A, we haven’t scored in four games but we’ll be back,” said Inzaghi.
The winner of this tie will play in May’s final against either Juventus or Fiorentina, who will face off in their first leg on Wednesday in Dusan Vlahovic’s first match in Florence since leaving for Turin in January.
In England, Middlesbrough produced another FA Cup shock to extend Tottenham’s long wait for silverware with a 1-0 win after extra time at the Riverside, while Manchester City eased into the last eight with a 2-0 victory at Peterborough.
Boro knocked out Manchester United at Old Trafford in the fourth round and the Championship side were worthy winners to condemn Spurs to another trophyless season.
Not since a League Cup win in 2008 have Tottenham lifted a trophy and another hugely disappointing defeat poses further questions over the future of manager Antonio Conte.
The Italian only took charge in November, but has already hinted he could walk away after a run of four defeats in the past six Premier League games.
“We have to give our best for the rest of the season then we will see what is our position and the evaluation of our season,” said Conte. “It is early to say something about this.”
Teenager Josh Coburn came off the bench to score the only goal with a thumping finish across Hugo Lloris in the 107th minute.
Earlier, Ukrainian international Oleksandr Zinchenko captained City as classy finishes from Riyad Mahrez and Jack Grealish sent the English champions through.
Fernandinho had initially been listed as City skipper for the tie, but the Brazilian, who signed from Ukrainian side Shakhtar Donetsk, offered the role to Zinchenko in a show of solidarity.
“It’s not an easy period for Oleks, his family, his country, but playing football is the best for him at the moment,” said City boss Pep Guardiola.
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