It was Spurs' first final in the competition and Pochettino spoke of his pride in his team for getting this far.
“It's so painful but at the same time we need to be calm in the way that we talk and analyse things,” he told Spurs’ official website.
“The season was fantastic and we need to feel proud, to finish in the top four and play in the Champions League final for the first time in the history of the club.”
Alli said he and his team-mates will use the result as motivation heading into next season as they look to build upon a fourth-place finish in the Premier League and another UEFA Champions League campaign.
“It’s been an amazing journey and I’d just like to say thanks to the fans for all their support,” he told the club's website.
“It’s a great learning curve and we have to keep this feeling inside of us, this hurt, and take it into next season.”
Spurs captain Hugo Lloris echoed that notion, insisting that they will come back stronger.
"First of all you need to digest the defeat," he said. "There are plenty of things to learn and I have no doubt the team will be stronger next season."