London Road was treated to the game's first shot after United's Craig Alcock let fly from distance, only to see his attempt skew just wide of the post.
The Blades looked to take the lead into the interval, but Jon Taylor had other ideas as he sent in a perfectly-timed corner, which allowed defender Brisley to guide a header past Mark Howard to make it 1-1.
Posh looked to build from the equaliser just before the half-time whistle, but Kyle Vassell somehow sent his shot straight at Howard, who saved comfortably.
United took control early in the second period, but failed to make their dominance pay as Baxter headed straight at the keeper from a few yards out.
Peterborough's Tyrone Barnett could have handed his side all three points with just under 20 minutes remaining with a shot from the six-yard box, which missed by a matter of inches.
Posh were left to rue the missed opportunity when Davies picked his spot from outside the box and drilled home past a helpless Ben Alnwick, leaving the final score at 2-1 to the visitors.