The Reds put in a solid performance to pick up their second league victory of the season and goals either side of half-time were enough to earn all three points.
Steven Fletcher's strike in the closing stages set up a nervy ending but Forest held out for the victory under the lights on Trentside.
There was one change to The Reds starting line-up from the goalless draw against Swansea on the weekend, with Joao Carvalho coming into the side to replace Joe Lolley who dropped to the bench.
Inside the first minute Forest carved out a great opportunity as Grabban did well to win the ball high up the pitch and poke it past Jordan Thorniley, but his cross was cleared, then Matty Cash arrived onto the ball but could only sky his effort high and wide of the target.
The visitors then had their first effort of the game on five minutes as Adam Reach latched onto a sloppy pass in the Forest midfield to drill an effort at goal which Costel Pantilimon was more than equal to.
On 15 they had a glorious chance to take the lead as Morgan Fox's delivery into the area was met by an unmarked Marco Matias eight yards from goal, but he could only head over the bar under no pressure from the Forest defence.
The Reds responded well to that big chance, with Cash and Saidy Janko – on his home debut – playing several teasing balls into the box which the Wednesday defence dealt with well. On 22 minutes it created a chance though as Janko's delivery was met on the edge of the box by a fine volley from Carvalho but he couldn't get his strike on target.
Wednesday forward Atdhe Nuhiu was a handful for the Forest defence and on 24 minutes he thought he had given the visitors the lead when he met a free kick into the area, only to see the offside flag raised.
At the other end, Grabban managed to get an effort away on 33 minutes despite being held in the area but could only divert wide and eight minutes later he wasn't to be denied as he bagged his first Reds goal.
Osborn played a fantastic ball into the box from a free kick on the right, and with the ball curling towards the far corner, Grabban was there to glance a header into the corner and past Cameron Dawson.
Forest took their lead into the interval and the visitors came out strongly at the start of the second half, forcing The Reds defence into making some key interceptions at the back to preserve their lead.
Then as the game passed the hour mark, Forest went back on the front foot and Osborn had a great chance to make it two on 62 minutes but saw his effort saved, then the midfielder linked up superbly with Carvalho to find the crucial second goal just a minute later.
The Portuguese played it into Osborn inside the area who played a neat back-heel back into Carvalho's path and, after a little shimmy, he curled a shot past Dawson and the covering defender into top corner.
Carvalho almost had his second and Forest's third on 76 minutes as he controlled Jack Robinson's long throw into the box, turned on a six-pence but volleyed straight at Dawson but there was a tense finale to come a Wednesday found a lifeline through sub Fletcher.
The Scotsman hit a shot from the edge of the box which took a massive deflection to wrong-foot Pantilimon and drop into the net. The Reds defence held firm though to see out the game and pick up their second league win of the season.
Forest: Pantilimon, Janko, Hefele, Fox (c), Robinson, Colback (Watson 81'), Guedioura, Cash (Dias 72'), Carvalho (Lolley 90'), Osborn, Grabban
Unused subs: Steele, Goncalves, Dawson, Darikwa
Goals: Grabban 41', Carvalho 63'
Bookings: Grabban 45', Robinson 78'
Sheffield Wednesday: Dawson, Baker, Thorniley (Pudil 73'), Lees (c), Fox, Onomah, Bannan, Reach, Joao (Fletcher 71'), Matias, Nuhiu
Unused subs: Wildsmith, Pelupessy, Kirby, Hector, Preston
Goal: Fletcher 88'
Referee: Tony Harrington
Attendance: 26,252 (1,995 away)