Ryan Yates struck just before the half-hour mark to put The Reds in front but to their credit, Boro fought back strongly and turned the game on its head before the interval with goals from Rudy Gestede and Lewis Wing.
It looked like Forest were about to slip to a disappointing defeat at the Riverside Stadium but Grabban produced a clever close-range finish for his 17th goal of the season to earn what could be a crucial point at the end of the campaign.
Sabri Lamouchi made one change from the 1-0 victory at Cardiff City last week as the injured Tiago Silva was replaced in the starting line-up by Alfa Semedo while John Bostock took a place on the bench.
Tobias Figueiredo had the clearest opportunity of the opening stages as the ball fell kindly to him just inside the Middlesbrough penalty area but the defender got his shot horribly wrong and it span well wide of Aynsley Pears' goal.
It had been a fairly tepid opening 29 minutes on Teesside with a lack of goalscoring chances for both sides but Yates was the man to break the deadlock. The midfielder picked the ball up on the edge of the box and drilled a low strike beyond Pears and into the corner of the net.
That goal seemed to stun Jonathan Woodgate's side into life and they almost found a response within a minute but Hayden Coulson's powerful drive flashed past Brice Samba's left-hand post and Wing had a decent opportunity soon after but his shot was superbly blocked by Figueiredo.
Gestede, Forest's nemesis for so many years, was the man to bring Boro level five minutes before the break. A free-kick to the back post was headed back across goal by Harold Moukoudi and Gestede arrived right on time to squeeze his header into the net from close range.
From looking fairly comfortable when they took the lead, Forest had been stunned and they found themselves behind at the break. Middlesbrough produced a swift counter-attack and it ended with Wing driving the ball beyond Samba despite the best efforts of The Reds' number one.
Forest were toiling as they tried to get level and Lamouchi made all three of his changes, bringing on Adama Diakhaby, Joao Carvalho and Bostock to try and get back into the game. The Reds had a couple of close-range chances as Figueiredo's header from a corner was blocked over the bar before Grabban poked a shot wide at the near post under pressure from Pears.
Resilience had been the key to so many of Forest's away games this season and so it was once again. Joe Lolley met Carvalho's corner on the edge of the area and as his effort span up into the air off a deflection, it fell to Grabban inside the six-yard box and with his back to goal, the striker delivered a clever finish to guide a spinning ball behind him and into the Middlesbrough net.
The Reds could even have won it in stoppage time but Diakhaby's firm strike was blocked away inside the penalty area and so the game ended level in the North East as Forest made it four games without defeat.
Forest: Samba; Cash, Worrall, Figueiredo, Ribeiro; Watson (c), Yates (Bostock 78'); Lolley, Semedo (Diakhaby 56'), Ameobi (Carvalho 69'); Grabban
Subs not used: Muric, Jenkinson, Dawson, Walker
Goals: Yates 29', Grabban 86'
Booking: Carvalho 90'
Middlesbrough: Pears; Howson (c), Shotton, Moukoudi, Johnson; Clayton, McNair, Wing (Saville 81'); Tavernier (Spence 87'), Coulson, Gestede
Subs not used: Mejias, Wood, Morrison, Nmecha, O'Neill
Goals: Gestede 40', Wing 44'
Booking: Shotton 75'
Referee: Darren England
Attendance: 18,884 (1,100 away)