After squandering numerous chances to win the game in normal time, United finally got the job done thanks to Marcus Rashford's latest match-winning strike with 107 minutes on the clock to seal a 3-2 aggregate win at Old Trafford.
"We did not start well because we did not capitalise on the goal," Carrick told BT Sport. "We created a few chances and didn't take them again - we thought it might be one of those nights. But we're through - that's what you have to do."
United could face one of Ajax, Celta Vigo or Lyon in the last four of the competition, with the draw due to take place on Friday morning.