Pablo Mastroeni, who took over as interim manager of Real Salt Lake after the sudden departure of Freddy Juarez in August, said he wasn't surprised to see his team pull off the comeback.
Perhaps he shouldn't have been, since it was the eighth second-half stoppage-time goal of 2021 for the club.
Those goals included Damir Kreilach's stoppage-time winner against Sporting on the final day of the regular season.
"The way things have gone this year, the way we got into the playoffs, the fight in these guys, the resilience ... I'd like to say I'm surprised but honestly when we scored that (first) goal I knew we were going to get a second one," Mastroeni said. "These guys believed they were going to get a second one."
Sporting's Johnny Russell opened the scoring in the 24th minute with a crisply taken penalty after Aaron Herrera tripped Gadi Kinda in the area.
Kansas City almost doubled their lead in the second half when Pablo Ruiz headed a free-kick from Kinda off the post.
The tenor changed, however, when Mastroeni brought in Anderson Julio, Justin Meram and Wood, and Julio grabbed the equalizer in the 72nd minute when he headed in a ball served up by Andrew Brody from just outside the box.
Finally Wood found the winner in the first minute of stoppage time, sliding forward to collect a low cross and firing it past Sporting keeper Tim Melia.
"It's amazing," Wood said. "We were down 1-0 but we kept going. I'm happy we got the goal and I'm happy I got to score the game-winning goal."
In Philadelphia, the Union reached their first Eastern Conference final with an emphatic shoot-out win in which Nashville failed to find the net in four attempts.
Union goalkeeper Andre Blake set the tone, saving the first two shots he faced in the decider with a dive right to deny Hany Mukhtar and a leap left to stop Anibal Godoy.
Nashville keeper Joe Willis saved a shot by Sergio Santos, but that was sandwiched between to made penalties for Philadelphia and when Nashville's Alex Muyl and Walker Zimmerman both fired wide the shootout, and Nashville's season, was over.
The teams had exchanged goals in the first half, with Mukhtar opening the scoring against the run of play in the 38th minute when he raced onto a cross from Eric Miller and headed home.
Daniel Gazdag grabbed the equalizer moments before halftime. Gazdag's initial shot off a corner kick was blocked but Gazdag collected his own rebound and fired past Willis.
In the Eastern Conference final, the Union will face the winner of Tuesday's conference semi-final between the New England Revolution and New York City FC.
New England enjoyed a first-round bye as top seeds, having won their first Supporters' Shield as the team topping the regular-season table.