The Blades were leading through Ched Evans' early strike but were reduced to 10 men after just half an hour when youngster Matthew Lowton was shown a straight red card.
And Cardiff, last season's beaten play-off finalists, dragged themselves level when Michael Chopra rounded United goalkeeper Steve Simonsen and teed up Bothroyd for the equaliser.
The hosts were simply glad to get back to playing football after a turbulent summer off the pitch - they were hit with a transfer embargo which was only lifted on Friday.
They made the better start and had the ball in the net after 13 minutes when Chopra burst through, but the striker was penalised for a foul on Johnny Ertl.
And 10 minutes later, against the run of play, United took the lead.
Evans could have been awarded a penalty when he was hauled down by Gabor Gypes, but the Wales international quickly got back to his feet when the ball broke to Leon Britton.
The summer signing from Swansea pinged the ball back across goal and Evans was on hand to turn it past Cardiff keeper David Marshall.
However, United were dealt a blow when Lowton, making only his third league appearance, was sent off by referee Peter Walton for a reckless high challenge on Kevin McNaughton.
The visitors made it to half-time without too much drama, and were relieved to see Mark Hudson's header from Peter Whittingham's corner fly wide after the break.
But City finally prised open the Blades rearguard on the hour mark when Simonsen came charging out of his goal to meet Chopra on the right of the penalty area.
Chopra's superb control of the high ball allowed him to jink around the diving keeper and feed Bothroyd, who was able to slot the ball into the net despite two defenders standing on the goal-line.
Simonsen made a better fist of saving at the feet of substitute Ross McCormack five minutes later after he was put through by Chopra.
The keeper also held on to a dangerous cross-shot from Darcy Blake, tipped another McCormack effort around the post and blocked Chris Burke's drive as Cardiff pressed.
He pulled off an even better save to deny Chopra from the edge of the area, diving at full length to palm away the striker's fierce shot, as the tiring 10 men held out for a point.