Boro's best chance fell to Jeremie Aliadiere whose low shot was saved by the legs of Ben Foster early on.
United boss Sir Alex Ferguson described the result as "a big step" towards retaining the Barclays Premier League title.
He told Sky Sports: "It's a big step forward - only four games left now, I keep saying you whittle the games away until there are none left.
"It was a very good performance....they all worked their socks off and once we got a hold of the ball we always looked a danger.
"Once we got in front, that was the key to it because they're a young side, Middlesbrough."
Giggs, named PFA player of the year on Sunday, and playing his 800th game for the club, added on Sky Sports: "We just wanted to get this one out of the way and we've done that and now we just want to concentrate on the remaining games."
Middlesbrough boss Gareth Southgate:
"We've gone into the last two games against teams in the Champions League semi-finals and were looking for a miracle. To have a chance you need to take the lead.
"But we did get stuck into them early on but it is very tough mentally when you fall behind because they are so good defensively.
"The second goal killed the game but I have to give credit to my players because they never gave up.
"We have got to put this game behind us and start preparing for our next match against Newcastle, which is a huge game for both sides."