United face Liverpool on Saturday in a mouthwatering fourth-round tie, having suffered three defeats and a draw in their last four visits.
Cole, though, is convinced United's current crop of stars can end that run this weekend.
"Look at the way we performed against Arsenal," said Cole, referring to Sunday's 2-1 Barclays Premier League win over the Gunners at the Emirates.
"If we go to Anfield and play like that, then we've got to be confident.
"Liverpool have been struggling for form recently, and if we play like we did at Arsenal then I can't see why we can't get a very positive result.
"We're long overdue a good result at Anfield. I thought we played well there earlier this season (in a 1-1 draw) and Liverpool aren't scoring too many goals at the moment.
"I think we can go there with a real sense of belief."
Cole knows that keeping quiet a partisan crowd inside Anfield is one of the keys to victory.
Speaking to manutd.com, Cole added: "Even if you've five or six thousand away fans there, you've still got to silence the 30-odd thousand Liverpool fans by keeping the ball and playing the way that United play.
"Liverpool will be very worried about us because they can see we're in good form. But we need to silence their crowd."
If United are going to go on to win the FA Cup this year they are certainly doing so the hard way after beating holders Manchester City in the third round.
But as Cole said: "If you're going to win the cup then you're going to have to knock over the tough teams.
"If you do that in the earlier rounds, hopefully it won't be as tough if you reach the final."