Daniel Farke's men began the campaign as favourites to reach the Premier League after missing out in the playoff promotion final to Southampton in May.
But the departure of Georginio Rutter to Brighton this week followed the loss of Crysencio Summerville and Archie Gray to Premier League clubs during the transfer window.
Leeds began their Championship campaign with draws against Portsmouth and West Brom, either side of a League Cup exit to Middlesbrough.
The visitors, though, steadied the ship in a Yorkshire derby at Hillsborough.
Brenden Aaronson swept in the opening goal midway through the first half.
Daniel James then rounded off a counter-attack from one end of the field to the other just two minutes into the second half.
Victory takes Leeds up to fourth in the table, just one point behind Burnley, Sunderland and Watford, who are the only sides to take a maximum six points from their opening two matches.
Wednesday are 13th are winning just one of their opening three games.
