After the shock elimination of Chelsea and Manchester City just over a week ago, United are the highest-ranked team left in the FA Cup.
So far the draw has been kind to Louis van Gaal's men
After beating Yeovil in the third round, they were pitted against Sky Bet League Two side Cambridge.
Although they were unable to win at the Abbey Stadium, United are big favourites to beat Cambridge in Tuesday's replay and set up a last-16 clash against Preston or Sheffield United.
Given that United are 10 points off first place in the Barclays Premier League, Shaw believes the FA Cup is the best chance he has of winning something in his first year at the club.
"The whole team look at it as a trophy we all want to win," the United left-back said.
"And especially now it's what you could say is the main priority because the league, I wouldn't say it's gone, but we're quite far away, so we're definitely looking to win the FA Cup."
With Chelsea and City both out of the FA Cup, the path to Wembley has been made a lot easier for United, who have not lifted the trophy for 11 years.
Wayne Rooney has been speaking to Shaw - and the other new signings at United - about his desire to end that long trophy drought.
And Shaw believes the draw has opened up nicely for him and his team-mates.
"Wazza hasn't won it and he's definitely going to want to win it this year," the England defender said.
"We've been quite lucky with the draw so far
We've picked up teams in the lower leagues so maybe it could be our year, especially now when you look at the teams that have been knocked out.
"There are not so many of the so-called big teams left so I think we have to take that as a positive.
"Hopefully we'll get past this one but you've seen from the past years that the big teams can get knocked out so we know it's not going to be easy and we need to be 100 per cent focused."
That focus has been sharpened by Bradford's win over Chelsea just over a week ago and Middlesbrough's victory at City.
"That's our warning for Tuesday," United boss Van Gaal said after his team's 3-1 win over Leicester on Saturday.
"Manchester United have not won it for 11 years so not always the best team win it
It's also the underdog."
Van Gaal almost has a clean bill of health for the replay.
Michael Carrick is unavailable, but Ashley Young could make his comeback from the hamstring injury he suffered on New Year's Day.
There is also a chance Van Gaal will rest David de Gea and hand Victor Valdes his debut in goal.
Cambridge boss Richard Money has likened his side's chances of success to winning EuroMillions, having already claimed a Las Vegas jackpot and a winning Lottery ticket.
The Us are 14th in League Two and the lowest ranked side remaining in the competition and Money played down his side's hopes of an upset immediately after the home contest
"If you go to Las Vegas and you play those fruit machines and you win the jackpot - that's probably what the draw was, in terms of Cambridge versus Man United," Money said.
"To get the replay probably means we bought a Lottery ticket and won
Now we've bought a EuroMillions ticket.
"The percentage of chance diminishes because of it, but, because you've got a ticket, you do have a chance
That's how I feel.
"To go there with no belief would be dangerous, but we're well aware of the task in front of us
"
Cambridge's finances are so tight that Money said his players would be charged for swapping shirts after the original match
The U's will earn 45 per cent of gate receipts after costs from the replay and while c hief executive Jez George does not know how much to expect, he is not about to start throwing money around
He said: "When you've worked here for eight and a half years with no money, run a youth scheme which hasn't had any funding and the struggle every month has been 'how are we going to pay the wages?'
"We're not going to be worrying about if it's going to be X hundred thousand or which part of a million.
"We're not going to spend it tomorrow, we're not going to waste it and it's not going to change how we do things at this football club
"
Source : PA
Source: PA