Richarlison came off the bench to equalise before setting up Dejan Kulusevski's 100th-minute winner, as Spurs came from behind to beat the Blades 2-1 in the latest-ever comeback in Premier League history.
Richarlison's first home League goal for the club came after he had revealed midweek he would seek "psychological help" following a turbulent five months away from the pitch.
"Not only me, everyone in the squad and club was very happy for him," Son said. "It was tough for him and we all hoped that this game would help his confidence. He changed the whole game. That’s what we were waiting for.
"Richy, [like] probably everyone, is not happy when not in the starting XI but we know how important it is to come on and change the game like he did or Ivan [Perisic], Brennan [Johnson], Emerson [Royal], Pierre [Hojbjerg]. Everyone did a really good job.
"It’s important to keep an eye on it. Everyone is very happy for him."
In scenes of wild jubilation after the final whistle, Son pushed Richarlison towards the South Stand at the Tottenham Hotspur Stadium to soak up the plaudits.
"This is part of [being a] family," the South Korean continued. "We always want to have a good time when we play with each other. Richy obviously had a very tough time, a tough season.
"But I was very, very happy. I was probably more happy than him. We need him as a team, he has really good quality but the confidence is massively different. For Richy, I just wanted to give him the big hug he deserved and he really showed his quality.
"Richy is a really strong guy, a good character and can always bounce back strong. But when you have a tough time you need good people around you.
“I always try to be a friend of him and if he needs anything then I can help him from my experience or my playing wise also. I think everyone is standing behind him helping. He did an amazing job for this club."
Over the last three or four seasons, Spurs' dressing room has often been fractured by rifts but Ange Postecoglou's new-look squad have shown real togetherness in an unbeaten start under the Australian.
"We are still on the way. Obviously big games, especially like Saturday, makes it more tighter and closer as a group," Son added.
"But I think obviously you can’t compare to the real family but we are getting really, really tight in the changing rooms. Everyone is working for each other, everyone is running for each other, everyone is fighting for each other. If someone comes off you give a hand off and everyone is happy to do that. That makes us really strong as a team and a group. We’re getting really close. We hope we can be even tighter than this."
Son's appointment as the new club captain has been a factor in unifying the dressing room, with Postecoglou saying the forward "transcends groups within the squad" after handing him the armband.
"I have good players and also amazing people around me, it makes my job really, really easy for me," Son said.
"I don’t try and be harder or difficult in a different way. I just want to be a good example by actions. I’m not the guy who is always talking and leading by words. I try to be a good example to everybody but having this group of players is making my job really, really easy."