Wednesday scored a second five minutes later when Chris Maguire drilled a low strike into the bottom corner.
Marvin Sordell pulled a goal back for Charlton a minute later after bursting forward and finishing past Chris Kirkland.
Nuhiu almost immediately restored his side's two-goal advantage, but poked his effort just wide of the post.
With three minutes of the first half left to play Charlton drew level, as Sordell was played through by Callum Harriott before firing past the keeper.
The Owls looked to quickly get back on level terms, but Jeremy Helan and Nuhiu both fired over the bar.
After the break Sordell came close to scoring a third, but his strike was kept out by Kirkland at his near post.
In the 63rd minute Sordell was able to complete his hat-trick with another clinical finish past Kirkland.
Wednesday pushed for a leveller late on, but the visitors held out to earn a vital three points in their battle against relegation.