Just one point separates the sides in the La Liga title race and they were the first two names out of the hat in the quarter-final draw in Nyon on Friday.
Atletico - three points behind leaders and city rivals Real - hold the advantage domestically and will meet Barca for the first time in European competition, with Valverde confident his side can get the better of the four-time champions.
"We've drawn Barca and will look to get through," he said of Atletico's first quarter-final since 1997.
"We are a team who bases its work on solidarity and persistence and we are uncomfortable rivals for any team."
The clubs have met on three previous occasions this season, each fixture ending all square.
The first leg will be played on April 1, with the second to follow on April 9.
