David Moyes' side could have taken control of Group A but saw Robin van Persie hit the post from the penalty spot, having hit the woodwork moments earlier, while Javier Hernandez somehow failed to find the net from six yards out with the goal at his mercy.
Marouane Fellaini saw red in the closing moments and United player-coach Giggs, who was making his 950th appearance for the club, felt they were unfortunate not to claim a rare victory in Spain.
Giggs told Sky Sports: "It was a good performance and I was just disappointed we didn't get the three points, we had the chances and on another day we would have won the game.
"It happens, you probably don't see Robin and Chicharito miss two chances like that in a game, but it happens, they are two great goalscorers, on another day they go in and we win the game.
"We had a great chance to open up a big lead, but we're still top, it's still in our hands and we always fancy ourselves at home.
"We showed some good stuff and on another day we would have won. I think we controlled the game and I think it was good preparation for a very good Arsenal team at the moment."