But they overcame Cardiff in the third round last month and victory at the Stadium Of Light would secure a place in the last eight.
Carew said: "In the last two years we went out of this competition in the second round and this is a trophy we can really go for and should be fighting for.
"Teams around us (in the Premier League) have reached quarter-finals, semi-finals and finals and there is no reason why we can't be there as well.
"It is important for Villa to do well in this competition having gone out of Europe. We are not in the Europa League anymore and we have to focus on this one.
"It is a chance to win a big trophy and we really have to go for it.
"As a big team, who are high up in the Premier League, we should be participating in quarter-finals, semi-finals and finals in this tournament - every year."
Villa boss Martin O'Neill reached the final of the competition three times in four years when manager of Leicester who lifted the trophy in 1996 and 2000 after being runners-up in 1999.
Now he wants Villa to achieve success but recognises Sunderland will provide a tough test.
O'Neill said: "I know during the last couple of years we have got knocked out early on which has been a big disappointment.
"Last season we were still heavily involved in the European scene and maybe the disappointment of going out might not have lasted that long.
"But this time around we don't have European football to go for and I would dearly love to do well in the competition if it is at all possible.
"The football club would be very keen to progress in it but we recognise that Sunderland, before Saturday's game at Birmingham, were having an excellent season and will prove a difficult test."
O'Neill is likely to make some changes particularly as his side are in the middle of a sequence of five games in 15 days.
Emile Heskey, Luke Young, Habib Beye, Nicky Shorey, Nigel Reo-Coker, Fabian Delph, Brad Guzan and Craig Gardner could all come into contention to start.
Sunderland v Aston Villa. Click here to bet.