When Rovers defender Dennis Souza received his second yellow card in the 58th minute, the stage was set for United to emerge victorious on Micky Adams' home bow at Bramall Lane.
The boyhood Blades supporter, 49, was last week handed a two-and-a-half-year deal as successor to Gary Speed.
Yet Billy Sharp, another lifelong United fan who failed to sparkle during his own time at Bramall Lane, opened the scoring in the 66th minute and rifled home a second from the penalty spot eight minutes later to put Doncaster 2-0 up.
Substitute Daniel Bogdanovic's close-range volley offered United hope in the 86th minute and Kozluk's late volleyed equaliser averted a fourth straight defeat.
While United were more physical than Rovers, the visitors knitted together some neat moves throughout.
United's first meaningful attack almost reaped dividends in the 10th minute when Kozluk's lofted pass allowed Ched Evans to break down the left channel.
The former Manchester City striker showed neat footwork to cut inside and drag a low right-foot wide of Neil Sullivan's near post.
John Oster then curled a teasing 25-yard free-kick narrowly over Steve Simonsen's bar and a vicious whipped cross from Rovers full-back James O'Connor caused panic in the home rearguard before the danger was cleared.
With Adams barking orders on the sidelines, United looked to break on the counter-attack in the 17th minute through Mark Yeates.
As the winger sought to break clear from halfway, he was cynically bodychecked by Souza, who earned himself a yellow card.
Sharp rifled home from close range in the 40th minute from free-kick but was stood in an offside position and the strike was rightly disallowed.
Sousa then committed another rash challenge on Evans in the 58th minute and his second booking left Rovers with a numerical disadvantage.
Yet Doncaster made light of the centre-back's absence by opening the scoring eight minutes later.
The impressive James Coppinger crossed low from the left flank and Sharp steadied himself before drilling a right-foot shot into the corner.
This time there was no stopping his celebrations and it got even better for the visitors with 24 minutes remaining.
Kozluk clumsily shoved Coppinger in the back and Sharp stepped up to fire home low to Steve Simonsen's right.
Bogdanovic fired home from 10 yards late on and there was drama still to come as Kozluk made amends for conceding the penalty by volleying home emphatically from 12 yards.