David Moyes' men lost 2-0 in their last 16 first leg match in Greece on Tuesday thanks to goals from Alejandro Dominguez and Joel Campbell, prompting widespread criticism over their lacklustre display.
But midfielder Carrick insisted he and his team-mates will respond strongly to the setback.
"We've set standards for ourselves as players over the years and this season hasn't gone so well," he said.
"We are getting shot at and criticised from all angles.
"Everyone is having a bit of a dig and enjoying it. We've got to take it on the chin and bounce back and I'm sure we will."
Although conceding the performance in Piraeus was way below United's best, Carrick remained positive ahead of the return leg in three weeks' time.
"It's a disappointing night," the England international added.
"We came here hoping to win and we haven't done that. We didn't react in the second half until it was too late.
"We didn't start the second half very well; we were a bit sloppy really and they scored the second goal, which put us on the back foot.
"But we're not out of the tie. We are still in with a chance going back to Old Trafford and we'll look forward to the second leg.
"Don't get me wrong, to score three goals in this competition is tough, but we've done it before and we believe we can do it again."
