Goals from Remo Freuler and Ruben Vargas mean Switzerland will face either England or Slovakia in a quarter-final match on 6 July.
Holders Italy have been knocked out of Euro 2024 after a 2-0 defeat to Switzerland in the last 16.
Italy defeated England on penalties to secure victory at the last men’s Euro final in 2021.
But Remo Freuler and Ruben Vargas’s goals secured a comfortable victory for the Swiss on Saturday.
It was the first victory for Switzerland over their southern neighbour in 31 years.
Luciano Spalletti’s side exiting the tournament means the Swiss team advance to the quarter-final of the European Championship for only the second time in their history.
They will face either England or Slovakia in Düsseldorf on 6 July.
The first goal came in the 37th minute from Freuler’s boot.
By the time Italy responded in the second, the Swiss defence were able to deal with it effectively.