Palace were reduced in number when Matthew Lawrence was sent off for deliberate handball six minutes into the second half.
However, the south London club would have snatched a point soon afterwards but for the agility of Lee Camp. Darren Ambrose's free-kick was heading for the top corner but the Forest keeper tipped the ball over the crossbar.
The win moved the Reds to within five points of second-placed West Brom who host Coventry on Wednesday night, while Palace remain in the relegation zone.
Both sides made one change to their starting line-up. James Perch suffered an ankle injury at the weekend so Luke Chambers was recalled in defence for Forest. On his return to the City Ground, Palace boss Paul Hart brought in Stern John for Nick Carle.
The former Forest boss suffered an early setback when Danny Butterfield pulled up with an injury after just five minutes. Sean Scannell replaced him.
In a first-half of very few chances, Palace keeper Julian Speroni was called into action soon afterwards to tip Anderson's long-range effort around a post and Robert Earnshaw shot into the side-netting after allowing the ball to run away from him.
At the other end, Camp pushed a fiercely-struck Ambrose free-kick away for a corner.
It looked like the first 45 minutes would finish goalless until Morgan belied his position as a centre-half by stroking the ball into the top corner from outside the area following good work down the left-hand side by Anderson.
Some sloppy Palace play almost saw Forest extend their lead at the start of the second half. A long ball forward sent Dexter Blackstock clear but rather than shoot he opted to pass to his striker partner Earnshaw whose effort was blocked.
Lawrence received his marching orders after he misjudged the bounce of the ball and prevented Earnshaw from racing through on goal.
But Palace responded well to the sending off and only a fine save from Camp to deny Ambrose kept Davies' side ahead.
Tyson replaced Earnshaw in the 80th minute and after little more than 60 seconds he settled the game. Camp collected a cross and immediately released Anderson down the left before Tyson converted his cross from six yards out.