The Dutch striker has enjoyed a fruitful first campaign after swapping the red of Arsenal for United, his league-leading 19 goals putting the Old Trafford side well on the way to regaining the title.
Van Persie was out of luck, though, in an intriguing first leg at the Bernabeu which ended in a 1-1 draw in the Spanish capital.
United, who led through Danny Welbeck's header only for former player Cristiano Ronaldo to equalise, were indebted to a series of fine saves from goalkeeper David de Gea.
Van Persie might have swung the tie in United's favour in the second half, having one fierce shot turned on to the bar by Real keeper Diego Lopez and a scuffed effort cleared off the line by Xabi Alonso soon after.
The Dutchman controlled the ball in the area but miscued his volley and Alonso was able to race back and hook the ball away with Lopez stranded.
"I had three [chances] in the second half, especially the one just after I hit the bar. That was a big chance. I should have put that one in. I just slightly mis-kicked it, didn't hit it properly," Van Persie said.
"With a bit of luck it could have gone in but they didn't and in the dying seconds the keeper made a really good save - I was counting on that one to go in."
In the frenetic dying minutes, Lopez palmed a Van Persie drive around the post.
Van Persie said the 1-1 draw was a "good result".
"We had a couple of good chances and I think if you'd said we would play out a 1-1 before the game, most of us would take it," the 29-year-old said.
"Away from home it's a good result. For the people who watched that game, they've seen chances on both sides and I think it was a great game to watch."
