Jack Lester's 54th-minute penalty earned victory at Highfield Road to boost Neil Warnock's side 48 hours after it was announced that Brown, suspended for this game, would be joining Tottenham during the transfer window.
But it was not easy against improving Coventry, who had edged into the top half of the table following two consecutive wins.
Eight minutes had elapsed before the first shot at goal came from Coventry's Patrick Suffo against his former club, the Cameroon international firing wide, and two minutes later he had an even better chance but shot over.
Phil Jagielka had United's first attempt, volleying over from a Stuart McCall cross, and after 22 minutes his team-mate Ashley Ward became the first player booked for a foul on Andrew Whing.
The yellow card was produced again three minutes later for Coventry's Michael Doyle following his challenge on Nick Montgomery, while Suffo was again off target with an angled volley on the half-hour mark.
A free-kick conceded by Youssef Safri produced the next opportunity for the Blades with Peter Ndlovu getting his head to Michael Tonge's delivery but the ball drifted wide, while Montgomery was cautioned for a foul on Yazid Mansouri.
The second half was only eight minutes old when the visitors took the lead.
Mo Konjic fouled Jack Lester in the area, earning a booking, and the former Nottingham Forest striker got up to place the penalty into the bottom right corner of the net.
Coventry responded quite well to going behind, although they were unable to create a clear-cut chance of an equaliser until the 74th minute when Calum Davenport got his head to Steve Staunton's free-kick but his effort flew wide.
Davenport had another sight of goal four minutes later but his header from a Mansouri cross was again off target, and four minutes from the end substitute Dele Adebola could have equalised from a Staunton corner but headed wide.
Ndlovu hit the woodwork in stoppage time as United pressed for a second goal but Lester's spot-kick proved enough to keep up the promotion challenge.