
Sevilla was on the verge of victory with leading 2-1 deep into added time, before Sergio Ramos equalised in the 93rd minute with the last kick of normal time, leaving the losers to taste a double dose of pain when Carvajal grabbed the late winner.
Marco Asensio gave Madrid the lead with a brilliant strike before Franco Vazquez equalised shortly before half-time.
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Ramos had looked set to be the villain of the piece when he conceded a penalty that Yevhen Konoplyanka's slickly converted in the 72nd-minute, which looked set to win it for Europa League champions Sevilla.
The Andalucians could have little to complain about over Carvajal's winner, as he flew into the box and clipped the ball past two defenders and the 'keeper with a lovely flick from the outside of his right boot. "The goal came from trying and trying again. Dani Carvajal's strike was spectacular, he deserves it more than anyone," Madrid's Lucas Vazquez said told UEFA after the game.
"Sergio Ramos is the first to never give up and he is always there in important moments. It is brilliant to have him. We have got a lot of depth and a side to take on all comers."
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Zinedine Zidane's Madrid were not at full strength, with neither Cristiano Ronaldo nor Gareth Bale, nor Toni Kroos, involved and both Luka Modric and Karim Benzema came off the bench in the second half at the Lerkendal Stadion. "We were trailing for all of the second half, but never give up and a goal in 90 minutes allowed us to become champions," Carvajal told UEFA after the game.
"Today we had a lot of players missing but we have a great squad of homegrown players."
Having just sold Jese Rodriguez to Paris Saint-Germain, Madrid started with Alvaro Morata in attack, back at the club from Juventus, but his performance was not upto the mark and he was substituted with French striker Karim Benzema just after the hour mark.
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