Barcelona were tested early at Spotify Camp Nou, with Newcastle starting fast and asking questions in the opening minutes. However, the hosts struck first through Raphinha, who finished a rapid counter-attack.
Newcastle responded when Anthony Elanga converted from close range, but Barca quickly regained control as Marc Bernal scored from a set-piece.
The visitors stayed in the contest, with Elanga grabbing his second after a defensive error, but Barcelona edged ahead again before half-time when Lamine Yamal converted a penalty.
Reflecting on the performance, Flick said: "It was a crazy game. We weren't sharp in the first half, we lost a lot of balls. But in the second half we controlled the game much better. What I liked most is that we spoke at half-time about how to play, and the players did it."
Barcelona took full control after the break. Fermin Lopez finished a flowing move before Robert Lewandowski struck twice, first with a header and then a composed low finish. Raphinha added a seventh after a defensive mistake to seal an emphatic night.
Flick praised Lewandowski's impact after choosing him ahead of Ferran Torres. "He told me he had the energy. He is experienced and in games like this he is one of the best. I'm very happy he scored two goals," the Barcelona boss said.
The German also provided an update on Joan Garcia and Eric Garcia, who both went off injured. "It doesn't look very serious, but we have to wait. The club will give more information."
Barcelona now move into the quarter-finals full of confidence, but Flick made it clear improvements are still needed as the competition intensifies.