The Welshman, who's played in 31 consecutive Manchester derbies, put the Reds in front on 17 minutes only to see former United striker Carlos Tevez turn round the deficit with a goal on either side of half-time.
"It was the perfect start," Giggs told MUTV. "You always want to go 1-0 up in a cup tie away from home. After that we probably weren't as crisp with our passing and kept playing the ball backwards. We let City back into it, really."
City's lifeline arrived courtesy of a dubious penalty. Rafael tugged at Craig Bellamy's shirt but contact appeared to be outside the box.
"We felt we were a little unlucky to go into the break at 1-1," Giggs admitted, "but we also knew we could play better.
"[The penalty decision] was a turning point. If we'd have gone in 1-0 up it might have been a different story. Sometimes you get decisions and sometimes you don't - you just have to get on with it."
He added: "Last season we comprehensively beat Derby County in the second leg of the semi so hopefully we can do that again. It's going to be tough - City are a good team - but there were enough signs - especially in the last half-hour - to suggest we're good enough to beat them."