Chris Hughton's side took the lead when Reading defender Tom Holmes forced the ball into the back of his own net, as he looked to prevent Glenn Murray from applying the vital touch to Sammy Ameobi's cross four minutes after half-time.
Meite's leveller did not come until the 81st minute but it was Veljko Paunovic's side who looked the more likely to grab a winner, as they spurned several late opportunities to secure a fourth-straight league victory.
Forest will have been happy to hang on for a point to bolster their efforts to pull away from the wrong end of the table.
The match burst into life when Gaetan Bong whipped in a cross from the left to perfectly pick out Murray.
His former Brighton team-mate conjured up a clever flick, but goalkeeper Rafael Cabral clawed the ball away.
Lucas Joao was allowed to charge right down the centre with the Forest defence backing off but Brice Samba was able to hold his low shot.
A flurry of action culminated in Joe Lolley cutting in from the left and lashing a shot just wide for Forest. The winger subsequently sustained what looked like a hamstring injury and had to be replaced by Anthony Knockaert in the 34th minute.
Forest had a let off when a clever corner routine ended with a great chance for Joao.
He failed to properly connect with a shot but, as the ball dribbled towards goal, the Reds defence was static and he almost caught up with the ball himself, before seeing it bounce off the foot of the post.
The second half began with Michael Olise curling a free-kick around the Forest wall – but Samba was well placed to save.
Forest responded immediately as Filip Krovinovic headed a Knockaert cross firmly into the side netting.
Forest did take the lead four minutes into the second half after James Garner had picked out Ameobi with a fine ball out to the right.
The winger beat Omar Richards before lashing a low ball across goal. Murray and defender Holmes stretched to connect and it was hard to judge who got the final touch.
George Puscas missed a great chance to draw Reading level almost immediately, as he fired over.
And goalkeeper Cabral had to make a smart save to deny Forest a second goal, after Ameobi had cut inside before hitting a powerful shot.
Ameobi flashed another shot wide as the home side continued to spurn chances.
Andy Yiadom was denied by the post as Reading pushed for an equaliser, while Joao sent a header over.
Joao had a role to play in the Reading equaliser as he picked out Miete, who beat Samba with a fine strike.
Holmes might have won it for the visitors, who mounted significant pressure, but he sent a free header wide at the far post as the spoils were shared.