The Netherlands international played an integral role in bringing the Premier League title to Old Trafford in his first season at the club after signing from top-flight rivals Arsenal, topping the league's scoring charts with 26 goals.
But United were eliminated from Europe's elite competition last year in a controversial 3-2 aggregate defeat to La Liga giants Real Madrid in the round of 16.
David Moyes' men topped their group this season and face Olympiacos for the right to reach the quarter-finals.
Van Persie admits any player would be "lucky" to win the Champions League and believes it is becoming harder to lift the iconic trophy.
Speaking to UEFA's official website, he said: "It would mean the world to me (to win the Champions League).
"Of course it's a trophy many players don't win. (If you win it) once, 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.
"Only one team can win it, and it's very hard. 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."
United have endured a difficult campaign in Moyes' first season in charge, and sit seventh in the Premier League table, 11 points behind leaders Arsenal.
And centre-back Nemanja Vidic believes the next month could define their season.
"We have to win regularly and in the next month we will know exactly where we are," he told The Manchester Evening News.
"By the end of January and start of February we will show what our position is and what our capability is in the Premier League.
"Then, of course, the Champions League starts up again. This is an important time for us and we all want to do well."
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