The Red Devils have qualified for the last 16 of the competition with two group matches to spare.
Having been drawn alongside Galatasaray, Braga and CFR Cluj, Carrick believes that United's campaign is yet to get fully underway.
"That is the big test," said Carrick. "That is when the competition really comes alive. We have done the first part. The second one is when we get the really big European nights and we go away to the best places in Europe and win games.
"We have managed to do that very well over the past few years. Hopefully we can get back to that."
United will take on Turkish outfit Galatasaray tomorrow night.