The Dutchman had three second-half opportunities, seeing two efforts well repelled by Real stopper Diego Lopez, but it was a scuffed effort which Xabi Alonso cleared off the line which most haunted van Persie.
"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 told Sky Sports.
"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.
"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.
"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."
Danny Welbeck's first-half header ensured the Reds have an away goal in the bank ahead of the return leg at Old Trafford, but van Persie warned against placing too much stock in its importance.
"At this level I don't really think that these kind of teams really care if they play home or away," he said. "We don't really care and they don't really care; it's just half-time now and we'll be ready for the second half."