United centre-back Matt Kilgallon was sent off in the 27th minute for a high tackle before Watson grabbed Wednesday's winner.
And in a bizarre second-half incident Wednesday's Jermaine Johnson was ordered back from the dressing room by referee Mike Dean and shown a red card two minutes after being substituted, for kicking a water bottle on his way down the tunnel.
Wednesday, 2-0 winners at Hillsborough last season, carved out the early chances in a scrappy opening start.
Midfielder James O'Connor tried his luck with a low 12-yard strike that was blocked by Kilgallon before Brian Howard blundered when he played a short back-pass to Ugo Ehiogu. Johnson nipped in but slammed the ball over the top
Wednesday centre-back Mark Beevers almost scored an own goal as he steamed in to cut out a clever David Cotterill cross before United nearly snatched the lead when Howard's low drive was turned past the post by Lee Grant after the ball had taken a wicked deflection off Marcus Tudgay.
The Bramall Lane side were reduced to ten men in the 27th minute when Kilgallon charged into Sean McAllister with his foot high catching the little midfielder in the back as referee Dean reached for a red card.
United manager Kevin Blackwell immediately brought off Cotterill for centre-back Chris Morgan who gifted Wednesday the only goal in the 35th minute.
Paddy Kenny charged off his line to deal with Beevers' hopeful ball into the box as Morgan nipped in to head clear and veteran midfielder Watson, spotting the United keeper off his line, coolly lobbed the ball into the empty net from 22 yards.
Wednesday almost snatched a second goal just before the break when Johnson's low cross from the left evaded the United back four and O'Connor toe-poked the ball a fraction wide of the post.
The Hillsborough side should have killed off ten-man United in the 55th minute when the evergreen Gary Speed clumsily upended O'Connor as he turned inside. However, Burton's penalty kick was poor and Kenny, diving to his right, was able to keep the ball out.
Johnson was red-carded after being substituted for Akpo Sodje and, in a nail-biting finale, United were awarded an indirect free-kick inside the area after Grant was punished for time-wasting but the Wednesday wall held firm.