United have not lifted the trophy since beating Millwall at the Millennium Stadium.
That record 10th triumph came so long ago only Ryan Giggs and Darren Fletcher of the present squad were involved, as was newly-installed Cardiff manager Ole Gunnar Solskjaer.
It was a point not lost on Sir Alex Ferguson prior to his retirement last summer, and one Cleverley is determined to put right.
"The old manager really instilled into us the fact a lot of players in the squad haven't won the FA Cup," said Cleverley.
"Growing up as a lad, the FA Cup final was an iconic thing.
"The run starts on Sunday and hopefully we can progress."
United entertain Swansea in the third round on Sunday, with the first leg of a Capital One Cup semi-final showdown with Sunderland following just 48 hours later.
At least it offers the Red Devils a diversion from their league form, which took another dip on Wednesday with that disappointing home defeat by Tottenham, their fourth loss at Old Trafford this season.
With Robin van Persie still to shrug off a thigh problem that has kept him out for six games and Wayne Rooney attempting to manage a groin injury, Moyes is hardly entering the second half of the campaign in a position of strength.
It merely brings added focus to United's work in the January transfer window, which Moyes has repeatedly stated is difficult to do any meaningful business in but others feel the club are likely to be active in.
"From all the noises coming out of the camp, I wouldn't be surprised to see a bit of activity," club ambassador Andy Cole told United Review.
"Our club, historically, doesn't do too much in January but we have a new manager with new ideas.
"I just wonder who will want to sell their better players halfway through a season. There are also inflated prices to consider as well.
"You get chairman bumping prices up because they know their players are in demand and other clubs will pay what they have to pay for what they need.
"Those chairman are in very powerful positions, especially if the players concerned are tied down to long-term contracts."