While Spain needed extra-time to see off a resurgent Croatia 5-3 in the last 16, Switzerland stunned world champions France 5-4 on penalties after a dramatic 3-3 draw in Bucharest.
This is the first meeting between Switzerland and Spain at the European Championship.
Their three previous meetings in a major tournament came in the 1966, 1994 and 2010 World Cups, with La Roja winning the first two and Switzerland triumphing last time out – albeit Spain went on to win the trophy despite that group-stage defeat.
However, that defeat in South Africa is Spain's only loss to Switzerland in 22 meetings in all competitions.
The teams met in October and November last year, in the Nations League group stage, with Spain winning 1-0 at home before drawing 1-1 on the road, and Luis Enrique is under no illusions as to the scale of test his team will have to pass if they are to face either Belgium or Italy in the last four.
"The reality is Switzerland have got through and nothing else matters," Spain's head coach told a news conference.
"The good thing for us is that both teams know each other very well. We competed recently in the Nations League.
"They're going to be a very tough team to face and I think for the spectator there might not be some big names, but they're a great group of players.
"They're a match for us in terms of the way they press, the way they attack, so it's going to be very difficult for us."
Spain are the first team in European Championship history to score five or more goals in consecutive games, having defeated Slovakia 5-0 in their final group match before edging Croatia in a thriller.
They had not scored more than four goals in any of their first 42 matches in the competition, though Switzerland have netted three times in each of their last two games, having never scored more than twice in any of their first 15 matches at the Euros.
"We need to be hungry again, greedy, to make it to the next round," Switzerland coach Vladimir Petkovic, who will be shorn of the suspended Granit Xhaka, told reporters.
"From this point on I can't say I'm satisfied and happy that we made it so far because, for me, the next step is always the most important.
"We want to succeed, make it to the next round. We know that we have to play against one of the strongest teams, Spain, one of the favourites, but we will try to take our chance and make it to the next round."