The visitors also threatened through Jose Baxter as the first half progressed, but the sides went into the interval goalless.
The red card handed the Blades the initiative and they took the lead in the 66th minute when Bob Harris fired Murphy's cutback into the top corner from the edge of the box.
Sheehan then went close with another free kick before Jonathan Stead struck the foot of the post with a header as Bradford pushed men forward in search of a leveller.
However, their urgency led to a mistake by Andrew Davies, who gifted the ball to Marc McNulty in the 83rd minute. The forward raced clear and kept his composure to finish as he sealed all three points for Nigel Clough's side.
Mark Yeates struck the bar late on as the home side refused to give in, but Darby's red card proved costly as the Bantams left empty-handed.