The Dons take on Dunfermline Athletic at East End Park in the third round tonight and McGinn feels it is about time the club from Pittodrie Stadium looked to end their 17-year silverware drought.
"It's a massive club with a good history," said the former Celtic attacker.
"There is no reason why we can't go on a run of victories and a good cup run.
"A club like this wants big days out at Hampden with good crowds following you.
"You want to play against bigger teams and going on a cup run gives you the chance to do that.
"These cup games are always tough, especially away from home against Dunfermline, who had a good result at the weekend (4-0 over Livingston) and will be on a high.
"They will be looking to take the scalp of an SPL team. But if we play to our capability there is no reason we can't come away with a win. You have to go in with the right attitude."