Morgan Gibbs-White scored the only goal of the game for Steve Cooper's side as they made it back-to-back wins across all competitions to move off the bottom of the table.
However, they needed a helping hand from Wilfried Zaha, who dragged a penalty wide of the target in the first half after winning the spot kick himself.
The result lifts Forest up to 18th in the table on a good day of results for them, with fellow strugglers West Ham United, Everton and Southampton all losing.
Cooper's men are now unbeaten in their last four home league games - a run which has provided eight of their 13 points in total this season.
The hosts did pose a threat in the first half as Brennan Johnson and Renan Lodi both came close, but Palace were handed a golden chance to take the lead shortly before half time when Zaha was dragged to the ground by Joe Worrall.
However, the in-form winger scuffed his spot kick wide of the target to hand Forest a reprieve - one they took full advantage of in the second half.
Gibbs-White fired home a rebound from close range to seemingly break the deadlock, although the goal was initially ruled out for offside.
A lengthy VAR check ultimately ruled that the summer signing was onside by the narrowest of margins, though, sparking a second round of celebrations from the home side.
Forest nerves will have still been jangling considering they have thrown away 12 points from winning positions this term while Palace have won more points from losing positions than any other side in the division.
However, the visitors could not find a response as they slipped to 11th in the table ahead of the winter break, leaving the Eagles still searching for their first-ever Premier League win over Forest in the process.