Goals from John Terry and Diego Costa sealed a 2-0 victory for Jose Mourinho's side at Wembley, as Pochettino's men struggled to break down their London rivals.
However, Pochettino felt there were positives to take and told Sky Sports: "We played how we needed to play always in our final.
"Firstly, congratulations to Chelsea and after my feeling is [that I am] proud of my players.
"I think we played very well but in the same way I'm disappointed for our supporters because we made a big effort.
"It was difficult but we need to look forward. We are proud because I think we played well.
"This group is strong, we showed character, personality, heart and this is important. I'm proud for them and we need to look forward."
Spurs came into the game off the back of a UEFA Europa League exit at the hands of Fiorentina on Thursday, yet Pochettino refused to use fatigue as an excuse.
"It is difficult but it is not an excuse, it is the truth. The reality for a lot of players is it was the first final and this is the first step for this group," he added.
"They are difficult to beat because it's Chelsea, one of the best teams in the world, and they have unbelievable players and I think maybe their players have played 10, 12 or 14 finals in their career.
"We need to improve but it's a good thing for the future."
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