The Portuguese was gutted after goals from Gareth Bale, Marcelo and Cristiano Ronaldo in extra-time gave Real Madrid European glory for the 10th time
Diego Godin's first-half header looked set to give Diego Simeone's men the victory before Sergio Ramos nodded home in second-half stoppage-time, taking the tie into a further 30 minutes.
Gareth Bale punished the Rojiblancos by heading Angel Di Maria's rebounded shot in before Marcelo and a Cristiano Ronaldo spot kick put the result beyond any doubt at Estadio da Luz.
Tiago was gutted with losing out on European glory but remained proud of Atletico, who beat arch-rivals Real and Barcelona to the Liga title this season, and says they will continue to give their all for the club into next term.
"We have given our all but, in the end, we had no strength," the Portuguese told La Sexta. "I'm very proud but very sad at the same time.
"We will continue giving our lives for this team."
Atletico remain without a European Cup or Champions League trophy, having been denied by Bayern Munich 40 years ago following a replay.
Contrastingly, Carlo Ancelotti's men wrapped up their 10th elite continental title - La Decima - with the 4-1 triumph.