New Charlton boss Karl Robinson watched on from the stands as Bauer turned home from close range in second half injury time.
Blades had looked to be claiming a fifth straight win after Mark Duffy finished off a clever set-piece routine in the first half.
Duffy later limped off and could be an injury doubt for Blades boss Chris Wilder ahead of the Christmas period.
A frustrated Wilder said: "I've let the players off in the last couple of games and people talk about stats, but the biggest one is putting the ball in the back of the net and we've dropped two big points here today.
"The amount of chances to put teams to bed over last couple of games has been frightening. We dominated start to finish against a good team with some Championship players, and for us not to go from one to two to three to four is disappointing.
"I'm looking for relentless and ruthlessness, we're a decent side but that ain't enough for me. I want us to be up there right now and that was a chance to put a marker down today, and say 'this is what we're about'."