The Owls, desperate to bounce back after five games without a win, survived a spell of early pressure before carving out a half chance, midfielder Graham Kavanagh picking out Wade Small, but his overhead kick lacked power as Peter Enckelman saved comfortably.
Cardiff, playing the slicker football, almost found a way through, 17-year-old Aaron Ramsey cutting in from the right to fire in a stinging shot which was blocked by the outstretched leg of centre-back Richard Wood.
Kavanagh drilled in a long-range effort, snuffed out by the Cardiff keeper, before Wednesday had a narrow escape, Ramsey firing in a low drive from 25 yards, but with Lee Grant beaten the ball cannoned off the foot of the post.
The Welsh side, whose impressive surge up the table ended with two successive defeats, pressed again, the evergreen Jimmy Floyd Hasselbaink picking out Paul Parry whose shot from 12 yards was cut out by Grant.
The Wednesday keeper pulled off a stunning save on 32 minutes to tip Hasselbaink's dipping effort from 25 yards over the top, before Peter Whittingham almost broke the deadlock with a curling shot from a similar distance, the ball flashing inches wide as Cardiff turned the screw.
But they were almost caught out on the break, Small finding space on the left to pick out Enoch Showunmi, who steered his glancing header wide of the target.
Small saw his shot deflected before Wednesday snatched the lead in the 41st minute, midfielder Adam Bolder picking out Showunmi, the lanky striker headed the ball into the path of Tudgay who drilled it past Enckelman from 14 yards.
The second half was a tense and tight affair, with neither side finding any sort of fluency, with Wednesday battling to hold on for a crucial victory.
Hasselbaink ballooned a shot high over the target and centre-back Darren Purse squandered a golden opportunity, heading a Whittingham corner over the target from five yards in the dying minutes.