Zac MacMath was called on for just one save while posting his second shutout of the season for Real Salt Lake (6-6-6, 24 points). The triumph halted the team's three-game winless stretch (0-2-1).
It is the 12th time expansion FC Austin (4-10-4, 16 points) have been blanked this season.
Wood's goal in the 32nd minute was set up by Jonathan Menendez, who delivered a cross in front of the net. Wood smacked the ball with his head and it sharply one-hopped into the left corner of the net for his second goal of the season.
Salt Lake put eight of 15 shots on target as Austin goalkeeper Brad Stuver made seven saves. Austin put just 1 of 10 shots on target.
Austin's chances of making a comeback were sabotaged when Julio Cascante was given a red card early in the second half, leaving his squad to play with just 10 men the remainder of the contest.
RSL's Justin Meram was trying to break toward the goal when Cascante fouled him and was given a yellow card in the 47th minute. But the officials reviewed the play and upgraded the card to red two minutes of clock time later when they determined Cascante denied an obvious goal-scoring opportunity.
Salt Lake had a chance to take a two-goal lead in the 80th minute when Maikel Chang sent a right-footer toward the top right corner of the net. However, Stuver got his left hand up to knock the ball away.
RSL had a big chance in the 20th minute but Toni Datkovic's close-range header was saved by Stuver.