"It was a really tough match; we had chances, they had chances," said Dest, who became the first American to play in the Clasico.
"It's too bad that we lost. They were maybe a little sharper. They finished their chances. I am pissed that we lost."
Dest, 19, returned to a more familiar right-back role against Madrid after starting the two previous Barca games at left-back, with Sergi Roberto dropping to the bench.
The USMNT defender was one of the few positives for Ronald Koeman's young side, with fellow teenagers Fati and Pedri also starting the match.
"It's my first Clasico and it's a nice game to play, of course," Dest added. "It's just too bad that we lost the game against Real Madrid.
"But we have to keep going. We must win the next couple of games now. We have Juventus next in the Champions League. We have to win that."
Dest said he didn't see the penalty decision which swung the game in the 63rd minute and left Barca boss Koeman fuming at the final whistle.
Referee Martinez Munuera eventually pointed to the spot after reviewing the pitch-side video monitor and deciding that Clement Lenglet's shirt pull on Ramos was a foul.
"For me, it's not a penalty," Koeman said in his postgame news conference. "I told the referee that I hope one day he can explain how VAR works here in Spain.
"We have played five games and VAR's only been used against Barca. It was not used for a penalty on Messi against Sevilla. It was not used for two Getafe fouls worthy of red cards in the first half last week.
"That was my question for the referee: Why is there only VAR against us?
"I have seen the replay. It's a foul by Ramos on Lenglet at first. Yes, Lenglet has hold of his shirt, but not much. If you've got a hold of the shirt like that, you fall forward, not backward."
Barca director Xavi Vilajoana went one step further than Koeman, writing on Twitter: "This is a f---ing scandal! Munuera, go f--- yourself!"
Despite the defeat, which was Barca's second in just five league games this season, Koeman saw reasons to be happy with his side's performance.
"We played well and created chances," he added. "We were also good in the second half. It's difficult to understand, we didn't deserve to lose the game. Until the penalty, we did well."