The 23-year-old striker, snapped up along with Rotherham's Deon Burton, hit the target in the 62nd minute after Richard Wood had broken the deadlock.
Paul Sturrock's shot-shy side had gone seven games without a goal before an unexpected 2-1 win at Burnley on Saturday.
Wednesday went ahead in the seventh minute after Burton caused problems in the Crewe defence, John Hills nipping in to finish with a stinging shot on the turn blocked by Ross Turnbull as he guarded his near post.
Richie Partridge, back in the side after on-loan Chris Eagles returned to Old Trafford, pumped over a corner from the left as centre-back Wood steamed to finish with a text-book header, his first of the season.
Burton O'Brien found a way though minutes later threading the ball to the unmarked Burton who pulled his low effort from 16 yards wide of the far post as the Crewe keeper charged off his line.
Luke Varney almost snatched an equaliser on 53 minutes after Gary Roberts picked out the little striker with Wednesday looking for an offside flag, but on-loan keeper Nicky Weaver was quickly down to block his shot at the foot of the post.
Burton was kept out by Turnbull, the Crewe keeper diving to his right to snatch the striker's glancing header from a Partridge cross before Tudgay opened his account in the 62nd minute.
The hard-working Hills fired in a low drive and the ball looped off Burton to Tudgay, who picked his spot heading into an empty net from six yards.
The unfortunate Chris McGready finished off a miserable afternoon for Crewe heading a cross from substitute Chris Brunt into his own net nine minutes from time.