Manchester United swept into an early two-goal lead at Old Trafford through Javie Hernandez and Wayne Rooney, 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.
They now face a tricky 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. The emotion and intensity of Tuesday night against Real Madrid took its toll on one or two of the players.
"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."
No date has been set for the replay, although should Chelsea progress in the Europa League this week, it will 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.
Rio Ferdinand admitted United had been punished for being careless.
He said: "We gave them the impetus to claw their way back into the game. We played some great football at times but were so careless at times. If you are careless you will get punished."