Wing came onto the field moments after Ashley Fletcher had fired home a 57th-minute equaliser for the Teessiders, cancelling out Sam Clucas' opener for Stoke.
The former Northern League player has developed a reputation for scoring long-range goals and having scored a cracker at Luton on the opening weekend of the season, he was at it again to extend Boro's winning run at the Riverside to three matches.
Picking up the ball at the heart of midfield, he drove towards the Stoke penalty area before drilling a superb finish into the bottom left-hand corner.
Woodgate said: "I was to-ing and fro-ing whether to put Wing in. In training yesterday, he smashed one or two like that in. But I thought I'd go with the experience of Adam Clayton, because of the young players that were playing.
"Lewis has got that in his locker though. He can hit balls constantly, but it's about doing it on the big stage. It's about transferring that from the training pitch on to stadium packed full of punters.
"I thought he was good when he came on, and I want my players to be disappointed when they're left out, I want them to be grumpy. I thought he responded really well, that's what I want."
As well as bringing on Wing, Woodgate also introduced Rudy Gestede shortly before the hour mark and the double-change was a major factor in Middlesbrough's success.
Having failed to threaten at all before the break, Boro created a succession of chances in the second half, with their victory lifting them six points clear of the relegation zone.
Woodgate said: "The first 45 minutes weren't great, I needed more from the players, they knew that.
"But I changed formation in the second half and we were miles better. We pinned their full-backs back, went man for man in midfield, looked more solid and had more of a foothold in the game.
"The fans were (restless) and I understand that. They expect us to steamroll teams, but it doesn't happen like that.
"Stoke played really well in the first half, Joe Allen was getting on the ball. But in the second half, the changes helped and the determination was a lot better."
The defeat leaves Stoke mired in the relegation zone and while their manager Michael O'Neill thought they deserved to take at least something from the game, he admits their position is becoming increasingly precarious.
O'Neill said: "I don't think it was a fair reflection of the game, I thought we were totally dominant up until we scored. The disappointment is we conceded straight after that and then conceded again.
"We have to look at why our performance level dipped and why we didn't create more opportunities than we did. Both of the goals were preventable.
"The team was bottom of the league when I came in, so I knew what I was walking into. I knew we were in a relegation fight.
"We have a squad people say should not be in this position, but the reality is this is the situation we're in. I believe we have enough not to be in this situation but every game is crucial now, as it was from the outset.
"I've only been in eight games, and in the eight games we've been ahead twice away from home and lost the games 2-1. In both of those games, a 10- or 12-minute period in the game seems to kill us.
"We have to start building some momentum and confidence going forward."