Barca won the round of 16 clash 2-0 thanks to goals from Ferran Torres and Lamine Yamal.
The win came a day after Real Madrid were knocked out at Albacete and Flick said: "There was a great atmosphere. Racing played very well and we got a good result. We're still in the Cup. We kept another clean sheet and advanced to the next round, that's the most important thing.
"For Racing, this is one of the most important matches in recent years. For us, too. It's important to know that you don't win because of your name, you win because of your attitude. We played a serious match.
"Our mentality was right. We did very well. Racing played a great match; maybe next year we'll see them in the Primera Division, we'd like that."
Flick also said of Segunda Division Racing: "They are brave and have nothing to lose. It's fantastic to watch them. The Copa is fantastic.
"I liked how they defended. They defended with everything they had. We saw their style of play. They press high. They did a good job."