Leeds deservedly took a 23rd-minute lead when Wood latched on to a fine cross from Gaetano Berardi before firing past Kieran Westwood from close range.
Wednesday then looked posied to equalise when the Owls were awarded a 53rd-minute penalty for a foul by Souleymane Doukara but Green produced a superb save to tip a low effort from Rhodes on to the post.
The victory took Leeds up to fourth and opened up a nine-point gap to those teams outside of the play-offs ahead of the rest of Saturday’s kick-offs.
Leeds began brightly at a packed Elland Road as a mistake by Vincent Sasso presented half an opportunity for Wood who dragged a difficult chance wide in the third minute.
Wednesday barely had a touch of the ball in the first ten minutes and it took until the 16th minute for the Owls to threaten when Adam Reach’s left-flank cross sailed through the Whites area and out of play.
Lets hit straight back with a similar attack from Hadi Sacko down the right wing but Sacko’s cross was eventually parried clear by Kieren Westwood.
But Wednesday then came the closest so far to opening the scoring in the 22nd-minute when Gaetano Berardi fouled Jordan Rhodes on the right wing and a Ross Wallace free-kick was headed narrowly wide by Sasso.
Rhodes then flashed an effort over from inside the area but Leeds had been the better team and the Whites sent Elland Road wild by taking a 24th-minute lead.
Doukara fed Gaetano Berardi on the left hand touchline and the full-back’s fine whipped cross sailed over Sasso and into the path of an unmarked Wood.
United’s top scorer then took one touch to bring the ball down and another to clinically finish to bag his 23rd goal of the campaign.
As Leeds pressed for an immediate second, ‘keeper Westwood then survived calls for a handball outside his area but Wednesday rallied and only a fine clearance from Pontus Jansson prevented Forestieri from striding clear.
After Eunan O’Kane gave away possession, Rhodes then looked to play in Forestieiri but the ball trickled wide.
But from a 33rd-minute Wednesday corner, Leeds messed up a fine chance to double their lead when countering three on two but Sacko failed to play in Wood or even Pablo Hernandez on the right flank.
From a Leeds corner, Wednesday countered in similar fashion in the 36th minute but the move ended with Forestieri skying the ball over the bar from 12 yards out.
United’s half was then slightly soured when Pontus Jansson picked up a 13th yellow card of the season for clattering Forestieiri, meaning just two more will result in a three-match ban.
And Wednesday caused an almighty scare in the 45th minute when a fizzed in cross by Reach was badly fumbled by Rob Green who survived as United hacked the ball clear.
Owls boss Carlos Carvalhal brought on Callum McManaman for Morgan Fox during the interval but Leeds went close to netting a second in the 49th minute when a superb pass from Liam Bridcutt found Wood who fired wide from outside the box.
Wednesday then survived big appeals for a Leeds penalty when Sam Hutchinson collided with Eunan O’Kane three minutes later in the penalty box.
And to the fury of Elland Road, Wednesday were then awarded a penalty of their own just two minutes later when Doukara was adjudged to have pulled back Hutchinson in the area from a Wednesday corner.
Rhodes stepped up to take the spot kick with Elland Road fearing the worst but Green produced a top drawer save to tip Rhodes’ low effort onto his left hand post and United hacked away the rebound for a corner amidst absolute scenes inside Elland Road.
Wednesday continued to press and first Forestieri and then Glen Loovens headed decent chances over before Almen Abdi wasted a free-kick which was fired over the bar.
Wednesday’s Ross Wallace was then booked for diving and trying to create a foul from Kyle Bartley who had his head in his hands thinking he was heading for a tenth yellow card of the season.
Leeds had not threatened for some time but did so in the 65th minute when a low cross from Luke Ayling was booted out to O’Kane who lashed a shot off target.
Carvalhal then made his final two changes in taking off Rhodes and Abdi for Steven Fletcher and Barry Bannan with 22 minutes left, with Rhodes jeered by Elland Road as he left the pitch.
Whites head coach Garry Monk then brought on Kemar Roofe for Sacko and Roofe almost made an immediate impression when attempting to squeeze the ball home from close range as Wednesday got in a pickle at the back.
Roofe’s effort was deflected behind for a corner which was cleared and Wednesday then forced two scrambles in the Whites area which Leeds survived in the 77th minute.
Amidst a spell of Owls pressure, Bannan then had a pop from 25 yards which sailed inches wide of Green’s left hand post.
Monk then took off O’Kane for Ronaldo Vieira with eight minutes left but hearts were in mouths in the 85th minute when Wednesday were awarded a free-kick out wide on the edge of the area for handball.
Leeds survived after Green was adjudged to have been fouled and Monk then made his final change in bringing on Stuart Dallas for Hernandez with four minutes left.
Wednesday huffed and puffed and Leeds then had to survive four minutes of added time.
Those four minutes were hectic and end to end but all that mattered for Leeds was that they survived which they did to bring an enormous roar inside Elland Road after nearly five minutes of added time.
Leeds United: Green, Ayling, Bartley, Jansson, Berardi, Bridcutt, O’Kane (Vieira 82), Sacko (Roofe 70), Hernandez (Dallas 86), Doukara, Wood. Subs not used: Silvestri, Cooper, Pedraza, Barrow.
Sheffield Wednesday: Westwood; Hunt, Sasso, Loovens, Fox (McManaman 45); Wallace, Hutchinson, Abdi (Bannan 68), Reach; Forestieri, Rhodes (Fletcher 68). Subs not used: Wildsmith, Winnall, Palmer, Semedo.
Referee: Michael Jones.