Grabban levelled with four minutes remaining despite claims from Boro that the striker had fouled goalkeeper Aynsley Pears.
The hosts had appeared on course to move out of the bottom three with a first win in 10 games after responding to Ryan Yates' 29th-minute opener with two goals in four minutes through Rudy Gestede and Lewis Wing.
But Grabban's intervention kept Jonathan Woodgate's side in the relegation zone and saw Forest edge to within eight points of second-place Leeds in the race for promotion to the Premier League.
Middlesbrough, who slipped into the bottom three over the weekend, made three changes to the side that turned in an improved display that still ended in defeat against Leeds last Wednesday, with striker Gestede a surprise inclusion in the absence of the ill Britt Assombalonga and injured Ashley Fletcher.
It was midfielder Wing who recorded the first shot on target from either side after 21 minutes but it was easy pickings for Forest goalkeeper Brice Samba who was on hand to comfortably gather.
But Boro soon found themselves behind. A routine throw was not dealt with, neither Alfa Semedo or Sammy Ameobi were closed down, and the latter's pass was dispatched past Pears by Yates from outside the area.
But Middlesbrough turned things around before the break after Hayden Coulson and Wing had already gone close.
With five minutes remaining of the half, Paddy McNair's deep free-kick was headed back across goal by Harold Moukoudi for Gestede to head down from under the bar for his first goal at the Riverside in more than two years
Pears then picked Coulson out with a long throw and the young winger ran deep into the half, rolling a pass to Wing who saw his shot hit the arm of the diving Samba before trickling over the line.
After a lack of chances at the start of the second half, Forest found their feet in a nervy ending for Middlesbrough.
Visiting defender Tobias Figueiredo headed over the bar from an unmarked position and then top scorer Grabban side-footed wide from close range.
But Grabban did level with four minutes to go. When Joe Lolley's volley from outside the area was deflected high into the area, the striker protected the ball from Pears with his back and left arm to flick a finish over the line.