England striker Danny Welbeck gave Alex Ferguson's side the lead at the Bernabeu before former Old Trafford star Cristiano Ronaldo levelled later in the first half.
"It's very open still," said Ferguson.
"We have a big job on our hands but it's within our grasp.
"This is a difficult place to come but the players have dug in and got a good result. We had great chances."
In the night's other last-16 tie, German champions Borussia Dortmund drew 2-2 at Shakhtar Donetsk in a game overshadowed by a horrifying plane crash at the Ukraine city's airport which killed five fans.
Real Madrid coach Jose Mourinho described his side's clash pre-match as one the "whole world was waiting for", it was the home side who came close early on with United keeper David De Gea, playing in the city of his birth, pushing a Fabio Coentrao drive onto the post.
But it was United who snatched the lead after 20 minutes when Welbeck shrugged off the attentions of Sergio Ramos to head in a Wayne Rooney corner.
Real were level after half an hour with Ronaldo living up the huge pre-match hype by scoring against his former club.
Ronaldo out-jumped Patrice Evra to send a powerful header off an Angel Di Maria left-wing cross past De Gea.
The Portuguese star's subdued celebrations reflected the warmth with which he still regards United where he spent six seasons.
Robin van Persie then had a shot saved by Diego Lopez with the ball coming back off the crossbar while, moments later, the Dutchman miscued a close-range effort which was cleared off the line by Xabi Alonso.
"Everything is very open for second leg," said Mourinho.
"They play very deep especially in second-half. They were waiting for a set piece to score. Even going there for second leg we can score goals."
In Donetsk, five fans of the Ukraine team, who had flown in for the match, were killed when their Antonov AN-24 plane, arriving from Odessa, was forced into an emergency landing.
There was a minute's silence before kick-off at the Donbass Arena.
Shakhtar took a 30th-minute lead when a Darijo Srna free-kick evaded the Dortmund defence before beating goalkeeper Roman Weidenfeller.
The German champions were level four minutes before the interval when Robert Lewandowski found the target after being set-up by Mario Goetze.
Shakhtar regained the lead after 68 minutes when substitute Douglas Costa scored with a left-foot drive but Dortmund finished on top and Mats Hummels levelled for 2-2 with three minutes left.
"It is important that we are going into the second leg level and a 0-0 would see us through," said Hummels.
"Certainly there were a few dangerous moments in defence, but against a such an attack-minded team you can never avoid this."