Luis Suarez and Lionel Messi struck before half-time in the semi-final first leg at the Vicente Calderon, and although Antoine Griezmann scored after the break, Barca held on for a valuable 2-1 win.
Simeone felt Atleti let themselves down in the first 45 minutes but remains confident that they could overturn the deficit at Camp Nou next week if they replicate their second-half showing.
"They started better, hurting us in behind the players going forward, and that's how the first goal came about," Atleti's head coach told a news conference.
"Then came the spectacular goal from Messi and that left us short of hope of producing a better first half.
"In the second half, we got closer to the team we've always been. We got ourselves close to the draw - I'm proud that we were able to lift ourselves again.
"I've always said that the Copa is strange: when it seems like you're far away, it goes from 0-2 to a possible 2-2 and that doesn't happen in La Liga, you don't see that kind of match.
"We will play for our chances and knowing that Barcelona will understand that the tie is not over."
Atletico host Leganes in LaLiga on Saturday before they travel to Catalonia for next Tuesday's second leg, and Simeone wants his players to continue the form they showed in the second half in order to build some momentum.
"What I said in the dressing-room [at half-time] stays there, but I'm very happy with the impression of the second half," he added. "We have to hope this continues and we'll be able to show it on Saturday."