Rangers took the lead 39 minutes into the game at Bramall Lane through Jay Simpson but the Blades drew level in first-half stoppage time when Richard Cresswell headed home from what was a contentious corner.
"We're in the hat after coming to a ground where many Premier League sides have floundered in the past," Hart said after the 1-1 draw.
"We fought really hard. We set up to be difficult to beat and we achieved that to some extent.
"I was disappointed to come in at half-time at 1-1 when the players are saying the ball went out of play before the corner was awarded. After that, closing down the cross was not there and the header was disappointing.
"But we withstood Sheffield United's pressure early in the second half and kept strong."