Simon faced criticism for a mistake that led to Croatia's opening goal in the last round, but he made amends with stops from Fabian Schar and Manuel Akanji in Friday's shootout win in Saint Petersburg.
"Football's been just today; we were worthy winners," Athletic Bilbao stopper Simon told reporters after the game.
"Now, just like we had to erase the memory of that mistake in the last match, it's time to quickly forget this win because we face a tough rival in the next round.
He added: "We have to go into the semifinal fresh, confident and with our heads high. We have to win the Euros now."
Denis Zakaria deflected home Jordi Alba's shot to give Spain an early lead, but Switzerland levelled after the break through Xherdan Shaqiri's third goal of the tournament following a defensive mix up.
Spain poured forward for the final 15 minutes and the entirety of extra time but were unable to beat the Swiss goalkeeper Yann Sommer, who made nine saves.
"Honestly, I'd have given it to Sommer," added Simon upon being named UEFA's official player of the match.
Spain coach Luis Enrique, who takes La Roja into the last four of a major tournament for the first time since 2012, was happy with Simon's response to his error against Croatia.
"It was the most relaxed penalty shootout I have experienced because everything had been worked on and I couldn't do anything else," he said in a news conference.
"What was going to be was going to be. From there, I had complete confidence in Unai and in the penalty takers. It didn't start off in the best way, but we came through.
"[Simon] had already demonstrated in the last game what he's made of, but I am so delighted for him. And I imagine his family and friends enjoyed him being the hero of the game."
Spain, who won the Euros in 2008 and 2012, will face Italy at Wembley for a place in the final.
"If there is a team prepared to overcome adversities, it is ours," Luis Enrique added. "We are one of the four favourites to win the competition on our own merits. I said from the start we were among the eight favourites to win it, without exaggerating. Now we're among the four best."
Switzerland, who stunned France on penalties to make it this far, see their best run at the European Championships come to an end after travelling more miles and playing more minutes than any other team.
Based in Italy for the competition, their run to the quarterfinals took in two games in Baku, one in Bucharest, one in Saint Petersburg and just one in Rome.
"I'm really proud of the team," goal scorer Shaqiri said. "Penalties are a bit 50-50. I was nervous watching the shootout. I think we just lacked a little bit of luck."