Switzerland controlled possession and pressed Italy onto the back foot but still Murat Yakin's team had to wait until the 24th minute before Breel Embolo's effort forced goalkeeper Gianluigi Donnarumma into action.
The Nati eventually broke the deadlock with 37 minutes played when Vargas' cutback pass found Remo Feulner, who made no mistake and tapped home past Donnarumma, reports Xinhua.
Donnarumma remained busy as he had to palm Fabian Rieder's dangerous free-kick around the woodwork just before the half time.
Switzerland continued on the front foot and doubled its advantage within seconds of the restart as Vargas finished the job into the top right corner following a counterattack.
Harmless Italy almost halved the deficit against the run of the flow after Switzerland defender Fabian Schar cleared the ball to the right post.
In the closing stage, Italy threw every man up front and came close in the 74th minute when Gianluca Scamacca rattled the woodwork.
However, Switzerland's victory was never in danger as Italy couldn't overcome goalkeeper Yann Sommer.
"We were also lucky. The almost own goal was a shock moment, but the team were great, and we were able to defend the result to the end. I hope this is not the end of the road for us and we can keep making history," said Switzerland coach Yakin.
"It hurts to go out like this, we apologise to everyone. They deserved it. We struggled the whole match. In the first half, we lost too many balls. It was tough, a very tough match to digest. In the first half we did badly because they always had the ball. We wanted to start better in the second half, but we conceded a goal straight away instead," said Italy goalkeeper Donnarumma.
Yakin's boys will face either England or Slovakia in the quarterfinals July 6.
(IANS)
Read more: Sports news
