The Reds remain 17th in the table, but they are now one point ahead of 18th-placed Luton Town, while the draw has left Wolves in 11th on 42 points from 31 matches.
Forest started well on home soil, with Wolves being forced to defend their own penalty box; an early strike came in from Ryan Yates during a period of pressure, but his effort was deflected wide of the post.
Wolves had a half-chance of their own in the 10th minute through Tommy Doyle, but his effort from the edge of the box was always rising harmlessly over the Forest crossbar.
Yates had another effort for Forest in the 12th minute after smart work from Callum Hudson-Odoi, but Wolves goalkeeper Jose Sa was on hand to make a smart save.
Hudson-Odoi had a chance of his own in the period that followed after Wolves failed to deal with a corner, but the former Chelsea forward missed the target, before Danilo had one deflected into the gloves of Sa.
Wolves had a half-chance of their own in the 18th minute, but Nelson Semedo's effort was blocked.
An open match was producing opportunities at both ends, with Giovanni Reyna missing the target for Forest after a mistake from Semedo, before Matheus Cunha fired over down the other end of the field.
Wolves came close again shortly after, with Murillo needing to make a goal-line clearance to deny Pablo Sarabia, before Joao Gomes missed the target from the next phase of play, with Forest wobbling in a defensive area.
The hosts continued to look dangerous going forward, and Neco Williams fired wide of the target in the latter stages of the first period after smart work from Reyna.
The breakthrough came from Wolves, with Cunha finding the back of the net in the 40th minute after driving into the Forest box and firing one into the far corner of the net.
Forest were not finished in the first half, though, with the score being levelled in stoppage time through Gibbs-White, who headed a corner from Reyna into the back of the net.
The hosts then took the lead in the 57th minute of the contest through Danilo, with the midfielder finishing into the back of the net from close range after smart work from Gibbs-White, who had driven into the Wolves box and refused to give up after initially losing the ball.
The drama continued, as Wolves levelled it up at 2-2 in the 62nd minute courtesy of Cunha's second of the match, with the attacker firing home from close range after Forest failed to deal with a corner into their box; there was a VAR check for a potential handball, but the goal was ultimately given.
Sa made a smart save to deny Hudson-Odoi, as Forest went in search of a winner on home soil, while Williams was also denied by the Wolves goalkeeper in the same period.
Wolves were also threatening, with Doyle striking over the crossbar in the final exchanges, but neither side could find a third, with the points being shared at the City Ground.