Mateta made the breakthrough in the 11th minute of the contest, but Wood's clever header just past the hour proved to be enough to secure a point for the home team.
Forest are 17th in the table, level on points with 18th-placed Luton Town, while the result has moved Palace onto 30 points from 29 games, leaving them in 14th, eight points clear of the bottom three.
Callum Hudson-Odoi was bright and inventive in the early stages of the contest, and he worked his way into a dangerous position in the second minute before delivering into the Palace box, but Eagles defender Tyrick Mitchell was on hand to make the clearance.
Palace managed to make the breakthrough in the 11th minute of the match, though, with Jefferson Lerma's through pass releasing Eberechi Eze, who in turn found Mateta, and the forward managed to fire into the back of the net to silence the home supporters.
Forest looked to hit back almost immediately through Hudson-Odoi, who was found after smart work from Morgan Gibbs-White, but Palace goalkeeper Dean Henderson was on hand to make the save.
Divock Origi was causing Palace problems at the City Ground with his movement, but he was finding it difficult to produce that piece of quality at the very last moment.
Ibrahim Sangare had a shooting chance for Forest in the 29th minute, but his effort from the edge of the box was always rising over the crossbar.
Forest goalkeeper Matz Sels made a smart save in the latter stages of the half to keep out a strike from Eze after Adam Wharton had found his teammate with a clever pass.
Lerma also tried his luck from distance late in the half, but he missed the crossbar, with Forest slightly fortunate to only be one goal behind at the interval.
Eze had another opportunity early in the second period, as Palace continued to push for the game's second goal, but the attacker just missed the target with a low strike.
Hudson-Odoi then curled one into the gloves of Henderson down the other end of the field.
Forest managed to level the scores in the 61st minute, though, with Wood expertly heading a brilliant delivery from Gibbs-White into the back of the net.
Forest goalkeeper Sels again made a smart save in the 69th minute to keep out a strike from Eze, who continued to look dangerous when he had possession in the final third of the field.
The home side were pushing for a second, with Gibbs-White firing over the crossbar after Henderson had kept out an effort from Giovanni Reyna.
Daniel Munoz also hit the post for Palace late on, but the points were ultimately shared at the City Ground, with Forest putting in a much-improved performance in the second period.