"The way we started the second half was a disaster," said Hart.
"We've gone 15 games without a clean sheet and you can see why. That is a side of our game that has to improve.
"It was a very spirited comeback but it was too little, too late."
Rangers' comeback, courtesy of Akos Buzsaky's penalty and Damion Stewart's header, set up a nervy finish but the hosts were unable to arrest their dismal FA Cup record.
They have not won a single tie in the competition since 2001 - a run now stretching 13 games - so it was little surprise to see United join a list of teams including Luton, Grimsby and Vauxhall Motors in sending them spinning out at the first hurdle.
Rangers' long-suffering fans were quick to show their frustration with a chorus of boos greeting Hart's first substitution, but the former Portsmouth boss is unfazed.
"I've not asked anyone if they are booing me or the team," he added.
"Ideally you'd send everyone away happy after a good performance. But people are entitled to their opinions."
The Blades were clinging on to Williamson's 19th-minute strike when substitute Ward scored with his first touch.
The striker, who is after a new contract at Bramall Lane, did his chances no harm when he tucked away Stephen Quinn's cross after 68 minutes.
And Ward turned provider seconds later when he charged down the right wing and set up the third for Cresswell.
"Good first touch, wasn't it?" said Blades boss Kevin Blackwell.
"We should have been two or three up by half-time so I'm glad we didn't have to rue those missed chances."
Buzsaky pulled one back from the spot after Derek Geary was adjudged to have handled - a "diabolical" decision according to Blackwell.
And hopes of a dramatic fightback were raised when Stewart headed in Buzsaky's free-kick with two minutes remaining.
But a point-blank save by Blades keeper Mark Bunn from Antonio German in stoppage time secured his side the dubious honour of a trip to Owen Coyle's Bolton later this month.