United return to European action next Tuesday when they travel to Olympiacos in the first leg of their last-16 clash.
After finding the going hard in the Barclays Premier League, United manager David Moyes will welcome a return to the European stage.
With United so far off the pace in the hunt for a top-four finish, winning the Champions League may be the club's best opportunity to qualify for the competition next year.
United did well to avoid the harder opponents when the first draw for the knockout phase was made, and, with the likes of Real Madrid, Barcelona and Bayern Munich in the tournament, Van Persie admits it will be hard for the Red Devils to reach the Lisbon final in May.
"Only one team can win it, and it's very hard," the Holland striker told Champions magazine.
"It's a trophy many players don't win; [you win it] once, if you're very lucky.
"You have a couple of players who've won it more than once but it's a very special trophy everyone wants to win every year.
"It seems to get harder every year to win it, because the teams are getting better and better. So it's always hard if you want something everyone wants."
The Champions League provided Moyes with a welcome break from domestic troubles earlier this season.
United took 14 points from a possible 18 in the group stages and they pulled off one of the results of the competition so far when they hammered Bundesliga high-flyers Bayer Leverkusen 5-0 in November.
Van Persie missed that game through injury, but he is now fit and almost back to his best.
And with a rejuvenated Wayne Rooney, who extended his contract with United on Friday, also back in the side, Olympiacos may be in for a long night.
"We cannot let Van Persie and Rooney play," the Greek club's director Pierre Issa told Olympiacos' website.
"We must be at our best. We have excellent defence in our team, and of course it will be a challenge for them to stop United's strikers, who are the best in Europe.
"The team will be very proud if they can stop Rooney and Van Persie scoring.
"It's a challenge to stop them. We will have to work hard."
All the omens look good for United ahead of Tuesday's game in the Greek capital.
Van Persie scored against Olympiacos for Arsenal in 2009 and United have won all four of their meetings with the Greek side.
That said, Olympiacos are enjoying a good season domestically under Michel.
The former Real Madrid player has managed to get the best out of veteran striker Javier Saviola this term and his team sit unbeaten atop the Greek Super League.
Olympiacos' main strength, however, has been in defence. The Greeks had conceded just nine goals in 25 games going into this weekend's match against OFI Crete.
One of the main reasons behind the club's defensive record has been the excellent form of Kostas Manolas - the centre-back who Moyes almost signed two years ago when he was in charge at Everton.