Tudgay bagged a deflected equaliser three minutes from time to deny Saints a first victory at St Mary's since September 30.
And Wednesday could even have won but for two brilliant stoppage-time saves from home goalkeeper Kelvin Davis.
Southampton needed a victory to avoid setting an unwanted record of their worst home start to a season since joining the Football League in 1920.
Jan Poortvliet's young side got off to the perfect start when Bradley Wright-Phillips put them in front with a superb 14th minute effort.
Wright-Phillips had the beating of Wednesday defender Frankie Simek on the left wing before cutting inside and hammering an unstoppable right-foot shot past Lee Grant and into the bottom left corner from 14 yards.
The goal followed up Wright-Phillips's double salvo in Saints' surprise 2-1 victory at Reading last month.
The former Manchester City star would have been celebrating another brace but for Grant's fine near-post save to deny him from an identical position two minutes later.
Wednesday did not get going in the first half and Leon Clarke's header over from Etienne Esajas' corner was the closest they came to testing Davis.
Wednesday boss Brian Laws clearly read the riot act at half-time and the visitors came out more fired up after the re-start, without putting Davis under any real pressure until the dying minutes.
Southampton should have been further in front but substitute Matthew Paterson fired into the side-netting from four yards when it looked easier to score his first ever league goal.
Paterson was made to pay for that miss when Tudgay took aim from 25 yards and his right-foot shot cannoned off defender Alex Pearce before nestling in the bottom corner of Davis' net.
Tudgay had earlier blazed into the stands after being played through by substitute Wade Small.
Wednesday almost grabbed a late winner when on-loan West Brom striker Bartosz Slusarski went through one-on-one with Davis only for the Southampton keeper to pull off a brilliant one-handed save and keep out the Pole's thunderous shot.