Blackman's goal preserved United's unbeaten home record after Crewe had come from two goals behind to lead in an entertaining match at Bramall Lane.
Shaun Miller edged the Blades ahead after three minutes with a fortuitous finish off his shoulder from Blackman's cross before Harry Maguire headed in unmarked on the half-hour mark from another Blackman delivery.
Crewe halved the deficit three minutes before the interval when defender Mark Ellis hooked the ball over Blades goalkeeper George Long after Alex captain Luke Murphy had rattled the crossbar from 25 yards.
And the visitors were level three minutes later when Mathias Pogba slotted his fifth league goal of the campaign from close range after goalkeeper Long had flapped at Murphy's free-kick.
Chuks Aneke side-footed in Byron Moore's cross two minutes into the second period to put Alex ahead, but Blackman levelled almost immediately after Miller's shot came back off the right-hand upright.
Lauri Dalla Valle almost put Crewe back in front in the 65th minute, but his finish flew the wrong side of the post while the lively Blackman was denied a penalty after he appeared to be barged in the box.