United swept into an early two-goal lead against Chelsea at Stamford Bridge, only to crumble after the interval.
Eden Hazard and Ramires pulled the visitors level and only a crucial late save from David de Gea kept United in the competition, even if they now face a trip to Stamford Bridge if they are to book a semi-final meeting with Manchester City at Wembley next month.
"We are lucky to still be in the FA Cup," said Ferguson.
"We ran out of legs. Our two full-backs, Rafael and Patrice Evra, were knackered and the two centre midfield players tired badly and we kept giving the ball away.
"I detected there were some signs of tiredness. You have to understand the problem is the emotion and intensity of Tuesday night had taken its toll on one or two of the players.
"Understandably I have no problems with that part but it made it a long day for us and they were by far the better team in the second half."
Typically given how much the focus had been on him, Wayne Rooney followed Javier Hernandez's early effort by curling home a free-kick which Petr Cech had expected to be cleared.
At that point United seemed to be cruising.
But a couple of missed opportunities gave the visitors' hope and Chelsea looked more likely winners at the end.
"We could have been four up in the first 25 minutes," said Ferguson.
"Wayne and Javier's goalscoring has been great this season.
"We needed that because by the end of the first half I detected our legs were starting to go.
"It was a very good save by David de Gea near the end. If they had scored then we were out there would have been no doubt about that."
No date has been set for the replay, although should Chelsea progress in the Europa League this week, it will inevitably mean the Londoners either facing four games in an eight or nine day period, or the postponement of Premier League games, one of which could be the Manchester derby.
The Football Association are expected to offer an update tomorrow, although Ferguson will be more concerned about Nani, who limped off with a hamstring injury just before the break.
"I don't think it is serious but he had to come off," said Ferguson.
"It was unfortunate because he was playing well."