The encounter kicked off at 1.25pm to mark United's 125th anniversary, but the occasion was soured 13 minutes into the contest when Henry fired Wolves ahead.
Henry, who scored in Tuesday's 2-1 defeat to Crawley Town, bagged his second goal in the space of a week by watching his teasing cross deceive goalkeeper Mark Howard to end the Blades' run of nine games without conceding.
The 24-year-old came within the width of the woodwork of doubling his side's advantage later in the half by crashing against the post from distance.
Nigel Clough's men would have restored parity shortly before the half-time break if not for the intervention of Richard Stearman, who produced a goal-line clearance to keep out a John Brayford effort.
It proved a decisive moment as the West Midlands outfit doubled their lead eight minutes after the restart through Edwards, who kept his cool with a first-time finish to convert a Matt Doherty pass.
However, two goals proved enough for Kenny Jackett's charges to move four points clear of Brentford.