Milan teenager Patrick Cutrone scored a sensational extra-time goal to dump Inter Milan out of the Italian Cup on Wednesday and book a semi-final showdown with Lazio.

The 19-year-old broke through after 109 minutes under driving rain at the San Siro after the match had ended 0-0 in regular time to set up a two-legged semi-final against the Romans who saw off Fiorentina 1-0 on Tuesday.

The victory was a welcome relief for new Milan coach Gennaro Gattuso who had likened the clash to “a World Cup final”.

“To think that yesterday I was in Primavera (Milan’s youth team), now cheering at San Siro with this magnificent crowd is something incredible,” said Cutrone who was booked for taking his jersey off in the celebrations after scoring.

“I think this victory was necessary for morale, the derby can’t be described as a normal game, but now we’re enjoying this victory and this shift and from tomorrow we will think about the challenge against Fiorentina.”

Milan went into the game in 11th place in Serie A, a massive 21 points behind league leaders Napoli, meaning the Italian Cup and the Europa League, where they have made the last 32, are their only avenues to silverware this season.

But it was a third consecutive defeat for Inter who are third in Serie A, five points off the leaders.

Luciano Spalletti’s Inter have lost steam following their strong start to the season, already needing penalties to get past third division Pordenone in the last round.

Goalkeeper Antonio Donnarumma made his senior debut for Milan as younger brother Gianluigi Donnarumma and Marco Storari were both injured. And the 27-year-old – who joined Milan last summer as part of the negotiations for the renewal of his brother’s contract – had a baptism of fire with the video assistant referee (VAR) coming to his rescue in the first half as he awkwardly turned the ball into his own net.

Inter had more possession in the first half, although Milan’s Giacomo Bonaventura had a chance to break the deadlock after 18 minutes but his header was tipped over the bar by Samir Handanovic.

Five minutes later Inter looked to have pulled ahead after Ivan Perisic knocked back a corner and Donnarumma turned it into his own net, but VAR confirmed Andrea Ranocchia was offside.

Inter captain Mauro Icardi’s header flew inches over as Milan’s Lucas Biglia blasted off target when unmarked on the edge of the box from a corner. And Joao Mario missed the chance to finish off an Icardi cross after an hour at the San Siro as Donnarumma saved, and the Milan goalie proved efficient, also clearing a Perisic strike and a Antonio Candreva close range effort.

But as the match headed into extra time it was to be Cutrone’s night as he broke through the Inter defence finishing off a Suso cross which left Handanovic with no chance to keep Milan on track for their sixth Italian Cup title and first since 2003.

The remaining quarter-finals between Napoli and Atalanta and Juventus at home to city rivals Torino will be played at the start of next month.