Having topped Group H, United were drawn against Group D runners-up Madrid in the knockout stages of Europe's elite competition.
The first leg of the tie comes next Wednesday at the Santiago Bernabeu, and Van Persie admits that he cannot wait to make the trip.
"The match against Madrid is the icing on the cake for every footballer," said Van Persie.
"It will be my first game at the Bernabeu - and I'm really looking forward to it. It will be a very nice challenge with the group of players we have.
"Manchester United are a big club and being in the last 16 is already beautiful even without the prospect of playing Real Madrid."
The Dutchman previously played against the La Liga champions when Arsenal defeated the Spanish outfit 1-0 at the 2008 Emirates Cup.