Both teams have enjoyed fine starts to the campaign and have now won two and drawn three of their five Premier League matches, while Brighton have also reached the last 16 of the Carabao Cup.
The hosts made a bright start as Simon Adingra caused problems, but his low, powerful cross from inside the penalty area could not find a team-mate.
Forest came into this match after an excellent 1-0 win at Liverpool and Nuno's side maintained their record of scoring first in each of their league matches this season.
Wood, who had a loan spell at Brighton in 2010-11, calmly sent home goalkeeper Bart Verbruggen the wrong way from the penalty spot.
Brighton gave Forest a warning when Georginio Rutter spun and shot wide from 20 yards, before Hinshelwood equalised with the Seagulls’ first effort on target.
That was quickly followed by Welbeck’s superb free-kick for his third goal of the season.
But Forest equalised midway through the second half and, following the three red cards, had a late chance to snatch the three points.
Sosa was clean through on goal in injury time, but instead of shooting he overhit a pass to Hudson-Odoi and the chance was wasted - though his blushes were somewhat spared when the flag was raised for offside.
Nevertheless, a point apiece was a fair conclusion to a hugely entertaining match.