Celtic took the lead in the 13th minute thanks to a clever lob from Scott McDonald and despite a host of chances, United had to wait until six minutes from time to secure a point thanks to Giggs' header.
"With so little time to go I suppose we'll settle for draw but with the way we played and our tempo in the second half, we probably deserved to win." Giggs told Sky Sports.
"It was a disappointing first half for us but the performance in the second half was much better. We came out and worked really hard and kept them pinned down.
"We knew chances would come. We knew we just had to keep the crosses and the one-twos going, keep the tempo up and it would come."
"It's always worth following up a shot like that with the ball moving all over the place," Giggs said.
"I just gambled and it fell right onto my head."