Home keeper Paddy Kenny was lucky not to concede a penalty and be sent off, while he was also struggling to stay on after suffering a knock, but had to battle on as Neil Warnock didn't have a goalkeeper on the bench.
David Unsworth's goal in injury time stole it for the South Yorkshire club to make it 3-2 and keep their automatic promotion hopes on course. Leeds's 0-0 draw with Plymouth - their 6th successive game without a win - leaves them 9 points behind the Blades with 4 games remaining.
At the bottom end of the table, Millwall failed to capitalise on Sheffield Wednesday's 2-0 defeat to Crewe. The Lions lost 1-0 at Derby, so Crewe overtake them and Brighton - who lost 2-0 to Southampton - into 22nd place. They're still 5 points behind Wednesday though. City, 19th ahead of Derby on goal difference, sit 11 points clear of Crewe and are almost mathematically impossible to be caught now.