The hosts, who fell to only their second home defeat of the season during Tuesday's nine-goal thriller against Fulham, dominated much of the match but fell behind to a Jeremie Boga effort in the first half.
The opening stages were bereft of clear chances for either side, with the best opportunity of the opening 20 minutes seeing Clarke nod wide after Enda Stevens and Jack O'Connell had combined down the left flank.
The hosts continued to have the better of things, with Stockdale denying a goalbound volley from Lundstram before Clarke again missed the target, but it was Birmingham who broke the deadlock against the run of play.
Jota's corner was only cleared as far as Boga, who cut inside before rifling his finish past Jamal Blackman, who had hitherto been untested.
The Blades struggled to respond to that setback and it wasn't until just over 20 minutes from time that either side had another notable chance when Lukasz Jutkiewicz saw an effort blocked while the keeper was off his line.
Sheffield United did level things up moments later, though, and it was the usual suspect as Clarke tucked a side-footed finish past an unsighted Stockdale for his 13th Championship goal of the season.
That proved to be the last meaningful sight of goal for either side as the spoils were shared in the Steel City, leaving United still third in the table and now seven points adrift of leaders Wolverhampton Wanderers.
Birmingham, meanwhile, move two points clear of the relegation zone having now won just one of their last seven outings in the league.
Sheffield United (3-5-2): Blackman; Carter-Vickers Carruthers, 68'), Wright, O'Connell; Basham, Lundstram, Duffy, Fleck, Stevens; Clarke, Sharp (Brooks, 68')
Birmingham (3-4-3): Stockdale; Roberts, Morrison, Dean; Nsue, Kieftenbeld, N'Doye, Grounds; Jota (Maghoma, 79'), Jutkiewicz (Gallagher, 91'), Boga (Adams, 87')