"We needed a fantastic performance tonight - and we got one," said Ferguson, who saw Paul Scholes net an early winner.
"It was a bit nervy in the second because they had to have a go at 1-0 down. But we got through that spell and in the last 15 minutes, I didn't think they made any chances.
"It was emotional at the end but you expect that at a semi-final. The important thing is that we are there and we deserve to be there."
Ferguson is hoping there will be an immediate pay-off in terms of United's performance against West Ham on Saturday.
Having seen their lead at the top of the Premier League whittled down to goal difference following the defeat at Chelsea, the Red Devils needed a lift and the Scot received the perfect assistance.
"It is a big boost for Saturday," he said.
"It won't do us any harm that's for sure. The energy in the team was excellent. The question is how do we get that back into them."