The Portuguese club lost their eighth European final in a row, this time on penalties, after being held to a 0-0 draw by Sevilla over 90 minutes and extra-time
The Portuguese club were held to a 0-0 stalemate by Unai Emery's men and both sides were unable to score in extra-time either, leading to spot kicks in Turin.
For the second season in a row - and the eighth European cup final in a row - Benfica were beaten at the last hurdle but the Argentine centre-back is determined to bounce back in 2014-15 on the European stage.
"It is not easy when you lose and even more so when we gave everything that we could," Garay, 27, told SIC.
"From tomorrow we have to think of next season to show how strong we are on the field, as we always have. Let us rise up and show that."
Benfica can console themselves having done the domestic double of the Primeira Liga and Taca da Liga, which is a far better silverware haul than when they finished runners up in all three competitions last season.
Sevilla, on the other hand, secured a spot in the Champions League with the Europa League final win having finished fifth in La Liga, which would only have sealed a place in the competition they have just won.