"We had all those chances and if we had scored them we might have made it to the final".
The manager praised Luis Suarez's performance.
Real are now nine points behind their fierce rivals in La Liga but do have a chance to close the gap on Saturday when the pair meet in the league.
Real Madrid boss Santiago Solari admitted he found the result hard to take given his side had been on the front foot for extended spells during the second leg.
The win was also historic as Barca became the first team in Copa history to reach six consecutive finals. They didn't take their chances in the first half but we took ours later.
Barcelona won the latest installment of the Clasico on Wednesday night. But this Real Madrid has something to say about it.
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The goal settled a fairly comprehensive 4-1 aggregate win for Barca, who will now face either Valencia or Real Betis in the final.
Another setback against Barcelona will practically end Madrid's title hopes in the league, leaving the Champions League as the only competition in which it can succeed in its first season without Cristiano Ronaldo in almost a decade.
A dominant yet wasteful Real ravaged their rivals in the first half while spurning a string of clear chances at the Santiago Bernabeu. "A week ago we talked about [Karim] Benzema being the best nine, Vinicius, Lucas [Vazquez]...we can not talk about Cristiano because he's not a Madrid player".
It didn't take long for Suarez to add another with a Panenka-style penalty in the 73rd after he was fouled by midfielder Casemiro inside the area.
Wednesday's match continued his surprisingly-poor run of having failed to score in any of his eight Copa del Rey fixtures against Madrid, but in the league it has been a different story with 18 goals and nine assists in just 24 meetings. "I don't think what we have achieved is something that will ever happen again in the Copa del Rey", Pique told reporters after the Clasico win. Messi and Dembele again combined but it was Raphael Varane, not Suarez, who'd end up bundling the cross into the net from close range.
But his selection in Madrid's starting line-up remains in doubt.