They reached the quarter-finals for the second time in their history with goals from Remo Freuler and Ruben Vargas capping an impressive performance in Berlin.
"There are only good sides at the Euros, we went unbeaten in the group stage and played well, and the Germany game showed we can go toe-to-toe with teams like that," Yakin told reporters.
"We sent out an important signal tonight with the game we played.
"We did not only club together and defend as a unit and sit back, we showed we could attack and dominate proceedings."
Switzerland's fans celebrated jubilantly with the players after their first win over Italy since 1993.
"It really sends a tingle down your spine, gives you goosebumps, it's hugely gratifying," continued Yakin.
"All this support shows we're on the right track and doing things the right way, we've earned the right to be here but we're not done yet."
Vargas set up the first goal for Freuler and then curled home a sumptuous second himself, having only been picked due to Silvan Widmer being suspended, but played on the opposite flank.
Yakin joked he would "take an aspirin" to resolve any selection headache in the quarter-finals, against either England or Slovakia, who meet on Sunday.
Vargas said Swiss captain Granit Xhaka had told him to grab the second at half-time and he obliged just 27 seconds into the second period.
"A little bit before the second half started, Granit said to me 'please score a goal'," recounted 25-year-old Augsburg winger Vargas.
"A few seconds later I found the ball at my feet and I did what Granit said and shot."
