After a tight opening to the tie Rangers went close after 14 minutes with Joe Cuff forcing goalkeeper Joe Green into action before Tom Morris cleared the rebound off the line.
Rangers were dealt a blow on 34 minutes when Daniel Burns was shown a straight red card when judged to be last man and bringing down striker Tom Pratt.
A penalty was also awarded and Paddy Lane put Hyde ahead from the penalty spot.
Minutes later the visitors almost doubled their lead as Lane squared for Pratt who this time failed to finish.
After the break Rangers were put under more pressure and were forced to make a clearance off the goal-line.
Stafford remained in the tie and Alex-Ray Harvey tested the defence with balls into the box.
With 20 minutes remaining Hyde had a second penalty shout but this time the referee waved away their appeals after Pratt claimed to have been fouled.
With minutes left on the clock Chinedu Uche sealed the points with his deflected shot finding the net.