Unai Simon was the hero as he saved penalties from Switzerland’s Manuel Akanji and Fabian Schar while Ruben Vargas skied over the bar after the match ended 1-1 across 120 minutes.
Enrique insisted he felt relaxed heading to penalties with Athletic Bilbao keeper Simon in goal, as 10-man Switzerland held on to force spotkicks.
“That was the most tranquil penalty shootout I’ve ever been through because we’d done all our homework, all our practice,” Enrique said at the post-game media conference
“There was nothing left for the staff and me to do. We all had huge faith in Unai, and after that all we could do is watch it and accept the result.”
He added: “People say penalties are a lottery but it’s not like that at all.
“Everything comes into play - the fitness, the ability to deal with pressure and the goalkeepers.
“It’s impossible to coach them but when you win it feels very good indeed.”
Spain moves into the Euro 2020 semi-finals where they will face Italy in London on Tuesday.
Earlier, Jordi Alba’s deflected early strike was cancelled out by Xherdan Shaqiri’s second-half strike.
The Swiss were reduced to 10 men in the 77th minute when Remo Freuler slid in dangerously which coach Vladimir Petkovic questioned.
“I question it a little bit, to be honest,” Petkovic said. “He maybe touched him a little.”