Roberto Mancini's side will face Stoke at Wembley after overcoming Manchester United in the semi-finals, with a 52nd minute Yaya Toure strike enough to knock their city rivals out.
City have failed to pick up silverware in 35 years but Barry feels that picking up one trophy will set the tone for a successful, more fruitful future at Eastlands.
He said: "The final is going to be as tough or even tougher. The pressure will be more on us, so we've got to learn from the United game and take that into the final."
"It is a massive moment for the club to reach the final. It has been a long wait.
"So it is important now that we win it. The next game is going to be just as tough, with the pressure maybe more on us.
"It is about winning that first trophy," he added. "It was mentioned at the beginning of the season.
"If we can win that trophy, then maybe more will follow. The first one is always the most difficult one."
"There were a few great individual performances but it was the perfect team performance.
"We all stuck together, as you have to do to beat Manchester United. Hopefully that will show we are together as a group."
Barry also refused to retaliate to Joey Barton's taunts earlier this month, in which he described the former Aston Villa man as a "teacher's pet."
However, the 30-year-old has refused to be drawn on the comments, stating that he would prefer to do his talking on the pitch.
He said: "You approach every game trying to do your best. I always do. If you get criticism, you get criticism.
"That's the game. You try and respond to it. I always give 100 per cent. If it's good enough, you get selected again. If it's not... I don't really want to speak about another professional, positive or negative."