Danny Welbeck's 20th minute header gave United the lead before Red Devils old boy Cristiano Ronaldo drew Real level in a pulsating first leg.
Alonso believes Real must adopt a positive approach to the return meeting at Old Trafford in three weeks' time, with a place in the quarter-finals at stake.
"I think that, after they scored, we knew we needed to be really careful as a second goal would have been a very bad result," Alonso, who cleared a Robin van Persie shot off the line in the second half, told Sky Sports 2. "Because of that, I think it's not bad at all.
"It could have been better because we had chances, and they had chances. I'm looking forward to the second leg, it's quite open the tie.
"They were really careful. Whenever they're playing away, they want to be really compact, really solid in defence. In the last 20 minutes they sat really deep and it was difficult for us to create chances.
"We knew that was going to happen at home and we are looking forward to the second leg at Old Trafford."
The former Liverpool midfielder continued: "We need to go for a win. We can't approach the second leg thinking about a draw.
"We want to go to Old Trafford to score goals, that's the best way to go through."