"I think overall we're pleased after going 2-1 down," Giggs told Sky Sports. "We want to win every game, but we hope that will be a point gained at the end of the season.
"We started the game really well and obviously got the goal, but Boro are a good team and it's always a difficult place to come."
Giggs admits the United squad are relishing the run-in, with the second leg of the UEFA Champions League quarter-final with Roma to come on Wednesday.
The Welsh winger added: "It's just about keeping your good form now, both as an individual and a team. You want to go into every game with confidence and knowing you can win every game.
"We know it's not going to be easy, but if we play to the best of our ability, then we'll have a chance.
"It's an exciting time, it's big game after big game and every game is like a little final in itself. You want to be involved in games like this and all the lads are looking forward to it."