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Luis Enrique's side ran out 3-1 winners on penalties after a 1-1 draw in 120 minutes of action.
Fabian Schar and Manuel Akanji had their kicks saved by Unai Simon, but Vargas saw his blazed over the bar.
The 22-year-old was in tears after Mikel Oyarzabal sealed Spain's victory - but Thiago couldn't let that stand.
Shaqiri and Thiago both consoling that kid:hearts: pic.twitter.com/Mtm9Yb79bv
- SIIIII SENORRRRRR (@tashaaa2000) July 2, 2021
Thiago, as well as Liverpool teammate Xherdan Shaqiri, gave Vargas words of encouragement before a huge embrace.
As well as Vargas, Thiago decided to have a few words with goalkeeper Yann Sommer before sharing a hug.
Fans have reacted to the gesture and they think Thiago has proven he's a genuine class act on and off the pitch.
Thiago went to comfort Vargas after missing his penalty and subsequently losing the penalty shootout.
Top sportsmanship from a top player :flag_es::raised_hands: pic.twitter.com/AyY37bG4cJ
- Row Z (@RowZMedia) July 2, 2021
Thiago consoling Vargas. That's my midfielder:thumbsup:
- M:flag_ps: (@MSleftfoot) July 2, 2021
Quality from Thiago consoling young Vargas after his penalty miss #ESPSUI
- Barry (@dcfc1928) July 2, 2021
Class from Thiago that, going over to console Vargas while the rest of the Spanish team were celebrating #sui #ESP
- Scouse Mick (@m_scouse) July 2, 2021
Thiago was the first over to Sommer and Vargas, can't buy class.
- SM:dizzy: (@SauceLikeTaki) July 2, 2021
Thiago going to see Vargas when he was upset though
- Bec :heartpulse: (@lfc_bec) July 2, 2021
Spain took the lead inside seven minutes after Jordi Alba's shot deflected in off Denis Zakaria.
The Borussia Monchengladbach man replaced Granit Xhaka, who was a hero against France in the last-16, and looked crestfallen.
Despite Spain having their chances, Switzerland grew into the game and got their equaliser in the second half through Shaqiri.
Switzerland went down to ten men not longer after thanks to Remo Freuler's controversial red card.
The Atalanta midfielder came in late on Gerard Moreno and saw referee Michael Oliver give him a straight red card.
VAR didn't save the 29-year-old and he trudged off the Krestovsky Stadium turf.
Sommer produced heroics in goal to keep the Swiss alive in extra-time, making vital saves from Moreno as well as Dani Olmo.
But Spain managed to advance and they'll face one of Belgium or Italy at Wembley on Tuesday evening.
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