Chances went begging at either end before Bannan struck to give the Owls a victory that strengthens their hold on a play-off spot.
Wednesday were without leading scorer Steven Fletcher, who was believed to be ruled out due to illness.
The hosts threatened again shortly afterwards with Bannan again the architect. His ball into the area was brought down by Reach who fired wide from a good position.
Kadeem Harris was next to threaten the City goal, cutting inside on the left-hand side of the area and firing in a low shot which was beaten away by Bentley.
Palmer also put a low shot on target which was easily dealt with.
Harris had a 20-yard effort saved by Bentley early in the second half and the keeper also had to be alert to block Reach's shot from inside the box.
Wednesday appealed in vain for a penalty when Rhodes went down under challenge from Williams.
As the home side continued to press, Bannan saw a volleyed attempt deflect off-target and then fired in a free-kick which forced a good save from Bentley at full stretch.
The winner came from the penalty spot five minutes from time, with Bannan sending Bentley the wrong way after Rowe was adjudged to have brought down Nuhiu.
O'Dowda's deflected shot was saved by Dawson, but otherwise, City never seriously threatened to force a late equaliser.