A brilliant Gustavo Hamer strike had initially cancelled out a Taiwo Awoniyi opener, but Wood came off the bench to pick out the bottom corner in the closing stages, firing Forest to a merited first win of the season.
Awoniyi's strike in last weekend's defeat to Arsenal proved to be nothing more than a mere consolation, but the Nigerian attacker would continue his hot streak in front of goal with just three minutes on the clock at the City Ground.
Blades defender Max Lowe swiped at thin air while attempting to clear the ball, allowing Brennan Johnson to surge into space in the right-hand side of the box, and the Welshman proceeded to cut back for Serge Aurier, whose cross was headed into the top corner by the in-form Awoniyi.
The 26-year-old's opener saw him net for the sixth Premier League game running - a feat only Erling Haaland and Christian Benteke have also achieved in the past 15 years - and Paul Heckingbottom's side showed little to suggest that they could peg their hosts back.
A speculative Vini Souza effort in the 34th minute - which was comfortably kept out by Matt Turner - was the Blades' best opening of the first half, during which both sides also had half-hearted penalty appeals waved away, while both Aurier and Morgan Gibbs-White fired over the top in the dying embers of the opening period.
However, Heckingbottom's half-time team talk invoked a fire within the Blades' bellies, as Sheffield United came out a different beast in the second half and drew themselves level in sensational fashion.
With 48 minutes gone, Forest could only clear a corner as far as new boy Hamer, who was left in acres of space on the edge of the box and bent an unstoppable curling effort into the top corner for a memorable debut goal.
Steve Cooper's side thought that they had risen back into the ascendancy just nine minutes later, but Awoniyi's chip over Wes Foderingham was chalked off for offside, before Heckingbottom was forced into a change as Souza was taken off for Chris Basham due to a knock.
Forest could have had few complaints about Hamer's leveller, having lost their way and then some early in the second half, but the hosts soon began to force the issue again and were denied another penalty in the 65th minute for Yasser Larouci's challenge on Aurier.
The resulting corner saw Foderingham thwart Willy Boly's effort on the stretch from close range, before desperate Sheffield United defending saw both Danilo and Anthony Elanga denied in quick succession in the 72nd minute.
It was truly anybody's game heading into the home straight, although Joe Worrall and Scott McKenna were indebted to Turner for denying Benie Traore one-on-one in the 80th minute, as the centre-back duo failed to deal with a long ball over the top.
However, the Tricky Trees quickly went down the other end, and another pinpoint Aurier cross did the damage, as the Ivorian's delivery was on the money for Wood to direct a header across goal and into the bottom corner with 89 minutes gone.
Forest navigated eight minutes of injury time to get off the mark for the new season at the second attempt, and Cooper's men have a full week to recover before they take on Manchester United next Saturday.
Meanwhile, Sheffield United - who remain without a point after two games of the new term - face the Red Devils' noisy neighbours and reigning champions Manchester City at Bramall Lane on August 27.