The Premier League leaders were seconds away from a third round exit at West Ham United before van Persie scored a stoppage-time equaliser to secure a 2-2 draw.
Giggs, who had just headed a seemingly easy chance over from close range, sent a 60-yard pass over the home defence and van Persie then beat West Ham goalkeeper Jussi Jaaskelainen from a position on the right of the penalty area.
"We missed some good chances, with headers over the bar and Shinji Kagawa had one blocked on the line, I thought it was not going to be our day," Ferguson admitted.
"But you have to say the pass from Ryan Giggs, then the first touch from Robin and his finish were world class."
The replay will add to an already-congested fixture list for United, who will resume Champions League action next month, but Ferguson made it clear the FA Cup was still a competition he wanted to win.
"We want to stay in it and have a go at it this year," he said. "We have had some terrible draws over the years and been knocked out early and we don't enjoy that."
Tom Cleverley gave United the lead midway through the first half but within four minutes James Collins had headed in an equaliser from a delivery from Joe Cole, who was making his first appearance in almost 10 years for West Ham, his first club, following his transfer from Liverpool.
Collins then headed in another accurate cross from Cole after the break and Sam Allardyce's side made it to the final minute with their lead still intact.
Ferguson was not pleased to have conceded two almost identical goals.
"For us to be losing 2-1 in that game is amazing with the way we played," he added.
"But when you play West Ham you have got to defend those crosses and set pieces, because they are brilliant at that -- the best team in the league -- but we did not do that.
"Twice in the same position, it was James Collins from a cross by Joe Cole and we have lost two goals. I could say it is a travesty, but they are good at it, but with some of the chances we missed, we should have been home safely."
Allardyce, who regards Ferguson as a friend and mentor, was disappointed not to be celebrating a famous victory.