Former Derby striker Tudgay hit the target in the 22nd minute with defender Spurr delivering the knock-out punch 11 minutes after the re-start.
Watford were handed a half-chance in the opening minutes when former Wednesday defender Leigh Bromby caused problems with a long throw into the goalmouth.
The ball was cleared to midfielder John Eustace who was lurking on the edge of the area and he drilled his fierce volley over the top.
Midfielder Jermaine Johnson, playing up front with Wednesday short of strikers, almost found a way through with a darting run as he swept past Bromby before full-back Lloyd Doyley nipped in to clear.
The Sheffield side took the lead in the 22nd minute with fortune on their side as Tudgay turned quickly in the box, his back-heel taking a deflection off Eustace as the ball rolled over the line with keeper Mart Poom wrong-footed on the line.
Watford should have been level in the 39th minute when Bromby rose to connect with Lee Williamson's precise corner but the big centre-back glanced his header wide from six yards.
James O'Connor had a stinging shot blocked minutes later and Johnson lashed the ball over the target from long range as Wednesday turned the screw.
They escaped a minute before the interval, Williamson curling his free-kick inches wide after Richard Wood had tripped Tamas Priskin on the edge of the area.
Lee Grant, a spectator for most of the game, foiled Jobi McAnuff minutes after the restart keeping out the midfielder's low strike from 18 yards with a one-handed save.
Wednesday grabbed a second in the 56th minute. Defender Spurr exchanged passes with Etienne Esajas before slotting the ball beyond Poom's reach from a tight angle.