The Red Devils earned a 1-1 draw in the first-leg last-16 clash at the Santiago Bernabeu, but could have carved out a lead had Real goalkeeper Diego Lopez not regularly thwarted the Dutchman.
And, after the game, Van Persie lamented his inability to hand Sir Alex Ferguson's a potentially decisive victory in the Spanish capital.
"I had three in the second half, especially the other when I hit the bar," the former Arsenal hitman told Sky Sports.
"That was a big chance, I should have put that in. I slightly mis-kicked it, I didn't hit it properly. So with a bit of luck, it could have gone in but it didn't.
"In the dying seconds, their keeper made a really good save because I was counting that one to go in."
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United goalkeeper David de Gea was also worked hard on his return to Spain, and Van Persie admitted either team could have come away with a win.
"Yes, for sure [it could have gone either way]," he added. "They had a couple of good chances and I think if you said it would finish 1-1 before the game, most of us would take it.
"Away from home, it's a good result but for the people who watched the game, they have seen chances on both sides and it was a great game to watch."
The result means that Real have to score when they travel to England, but the 29-year-old doesn't believe that fact counts for much.
When asked whether United have the advantage, he said: "Well slightly. At this level I don't really think these kind of teams care if they play at home or away.
"We don't really care, they don't really care and it is only half-time now. We will be ready for the second half."
Goalscorer Danny Welbeck also expressed delight after netting at a stadium he described as "magnificent", and praised his team-mates for not being daunted by the occasion.
"I knew where Wayne [Rooney] was going to put it and thankfully I put into into the back of the net," he said.
"You come to the Bernabeu and it's a magnificent stadium, but we came to play the game and not just the occasion."