The last-16 Champions League tie is balanced at 1-1 following Wednesday evening's first leg at the Santiago Bernabéu when Danny Welbeck's opener was cancelled out by Cristiano Ronaldo's equaliser later in the first half.
On the away goals rule this means a scoreless draw at Old Trafford would be enough for United to knock the nine-times European champions out of the competition.
But Giggs, asked if the tie was still 50-50, said: "Yeah, I think it is. You have got to recognise the quality Real Madrid have got. They won't be fazed by coming to Old Trafford and it is up to us to put pressure on them. The fans will be behind us and we'll need to put in a top performance like tonight [Wednesday]."
Despite Real having 28 shots at David de Gea's goal they managed to score just the once. "We have defended really well over the last few weeks," Giggs said. "We had clean sheets against Fulham and Everton and we defended really well both as a team and individually against Madrid. David has had to make some great saves and it was good for him to come back to Madrid and put in a big performance."
Michael Carrick and Phil Jones agreed with Giggs that the second leg on 5 March will be tight. "We can be relatively satisfied with the result," Carrick said. "With the chances we had we feel we left a goal out there but they have had opportunities as well. It was quite an entertaining game in that respect.
"We still feel that with another goal we would be over the moon. A score draw is still a good result. We are not getting too down about things. You have to respect them. They have so much talent and they are going to cause you problems.
"The disappointment was the goal. It was so simple. A cross and a free header. But we have seen Cristiano do that many times. Unfortunately he has done it again. They had other opportunities and some of the chances we had, it was one of those games."