Exeter, who have spent most of their 101-year history at the bottom of the Football League, recorded their greatest ever moment at Old Trafford by defying United, FA Cup holders and winners of the competition a record 11 times.
Ferguson said he regretted his decision to start with a very experimental second-string side.
"I apologise to the fans because they didn't deserve that, they deserve more than that," he told MUTV.
"The quality was poor, the quality that's here every week wasn't there. We got into great scoring opportunities, but it was Exeter's day and there's nothing we can do about that.
"We're all disappointed, and that's why I say to the fans 'I apologise'. In the 18 years I've been here, that's without a doubt the worst performance in the FA Cup.
"We'll do better in the replay. We're still in the Cup, that's the one great thing about it. There's some notable teams out of the FA Cup today and we can thank our lucky stars we're still in there.
"In the replay I'll play a much stronger team and hopefully get through." Exeter boss Alex Inglethorpe was delighted with a result that will bring in welcome funds from a money-spinning replay at St James' Park on January 18.
"It's very, very difficult to pick out heroes today. Everyone has played their part," he told Sky Sports News.
"I'm immensely proud of them and I think there's a lot of people in Exeter who will echo that."
Looking back at the Devon club's struggles in recent years, after being relegated from the fourth division in 2003, Inglethorpe said: "It is a fantastic story for a club that was nearly out of business a couple of years ago to rejuvenate itself.
"The fans have taken over the running of the club and they've gone from strength to strength."
Asked whether Exeter could pull off one of the Cup's biggest all-time shocks by toppling United in the replay, he said: "Why not?
"You've got to go into games believing you can win or there's no point turning up -- we may as well have sat with the crowd today," Inglethorpe said.
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