United, who won the league in 2010-11, finished second to Manchester City on goal difference in the league last season.
A last-gasp 3-2 victory for Roberto Mancini's side over Queens Park Rangers on the final day of the campaign broke United's heart, but Carrick insists that they have got over their disappointment.
"What happened takes some getting over," said the England international. "It's not easy. It was a huge blow, the way it happened.
"Nobody could have predicted that and we never thought it was going to happen. But we have to get over it now. We've had setbacks in the past and we have to move on.
"That's the nature of the club and the manager (Sir Alex Ferguson). He'll be driving us forward to achieve something this season."
Carrick further stressed that United will come back stronger this term.
"As a club we do tend to bounce back," claimed the 31-year-old. "If we have a defeat or setback, we tend to come back stronger.
"We can't forget how close we were. It wasn't a disaster in terms of how the season went because we had a good season.
"We'll bounce back. We're strong this year and we're hungry, not that we wouldn't have been hungry anyway. It will be a good championship."
United travel to Everton for their opening game of the Premier League campaign on Monday.