The Ukrainians had clearly studied the tape of Leeds' defeat at the hands of Sunderland because they came with the same tactics of getting ten men back with only the diminutive Irakli Modebadze up front.
And once again Leeds found it difficult to break down.
However, their lack of goals owed as much to poor finishing as to a failure to create chances with Jonathan Woodgate and Eirik Bakke heading over from close range.
The visitors were hoping to grab an away goal on the break and almost succeeded midway through the first half when Fabior Vasconcelos cracked a low drive but Paul Robinson got down smartly to gather it in.
Smith was pushed further forward in the second half and almost broke the deadlock ten minutes after the break but his header was clawed away by Fabien Barthez lookalike Andriy Glushchenko.
As Leeds increased the pace and pressure, Harry Kewell produced a fierce drive that was blocked by the midrif of Denys Smirnov.
Terry Venables made a double switch in the 65th minute, bringing on Steve McPhail to prevent Dacourt getting a second yellow card, and to roars of approval from the Leeds fans, giving Bridges his first appearance in 23 months.
With the goal looking as though it would never come, Leeds finally made the breakthrough with ten minutes to go - Bridges chasing Kelly's lobbed pass and pulling it back for Smith to sidefoot home.
Lucas Radebe, who had been outstanding all night, almost rounded it off but his fierce injury-time volley from a corner was pushed away by the keeper.
Leeds: Robinson, Kelly, Radebe, Woodgate, Harte, Bakke, Dacourt, Bowyer, Kewell, Smith, Viduka. Subs: Martyn, Bridges, McPhail, Mills, Okon, Burns, McMaster.
Metalurg Zaporizhzhya: Glushchenko, Valuta, Raty, Klyuchyk, Visevic, Dodic, Vasconcelos, Milosavijevic, Smirnov, Akopyan, Modebadze. Subs: Zolna, Hansen, Zayats, Brdanin, Rodri, Lapko, Savinov.
Referee: M Vuorela (Finland)