"We've got to maintain a balance in life and even for him, he needs to understand his football is still good. He's a fantastic footballer," Postecoglou told BBC Sport.
"He's got so much to give and that can help ease the burden he's feeling in other parts of his life. I'm really pleased for him."
Until Richarlison's intervention, Spurs' unbeaten Premier League start under Postecoglou looked to be coming to an end thanks to a Gustavo Hamer strike that crept in off a post.
Sheffield United had seemingly engineered a classic smash-and-grab away victory after soaking up pressure for long periods, with goalkeeper Wes Foderingham making a number of fine saves.
But Richarlison glanced home fellow substitute Ivan Perisic's corner before Kulusevski left the gallant visitors, who had Oli McBurnie sent off at the end, broken-hearted.
Having trailed until the 98th minute, Tottenham's turnaround was the latest winning comeback in Premier League history.
"There's some real resilience and spirit within that group already, which is great to see because they are a relatively new group," said Postecoglou.
"You get tested on days like this. It helps fuel belief for us to grow. Our supporters have had an unforgettable day and all those things help in what we are trying to do."
Spurs find another way to win
Spurs have been hailed for their fine start to life under Postecoglou - and to life without Harry Kane - with their vibrant performances helping the Australian win the Premier League manager of the month award at his first attempt.
But, of their four wins so far, this victory arguably might please him the most given the way Spurs had to battle and persevere to eke it out.
Most of their opponents this season had been happy to go toe-to-toe with Postecoglou's famed 'Ange ball' philosophy, but that was always unlikely with Sheffield United.
Paul Heckingbottom's side came with a clear gameplan to stifle Spurs, and it succeeded for much of the afternoon, although they still needed Foderingham to be at his best.
Brennan Johnson thought he had equalised with virtually his first touch in a Spurs shirt, but the offside flag denied the deadline-day signing, leaving the stage clear for his replacement to steal the show.
Richarlison has endured a difficult time since leaving Everton for north London last summer, often employed as a substitute and scoring only once in the league before his header here.
But he - and Postecoglou - will hope the honesty he displayed off the pitch this week and the impact he made on it against United can be the turning point of his Spurs career.
A fourth successive victory means Spurs have their best start to a season for 57 years and keeps them second in the table before Postecoglou's first taste of a north London derby against Arsenal at Emirates Stadium next weekend.
The winless United, meanwhile, remain on one point from five games despite redoubtable efforts against both Spurs and Manchester City - with all four of their losses coming by a single goal.
They will be without McBurnie for at least one match after he was handed a second booking not long after Tottenham's winning goal.
"It's sickening for everyone involved in the dressing room - to come so close to three points, then thinking you're going to have settle for one, and come away with nothing," said Heckingbottom.
"But that doesn't alter the performance. Right now we have to be positive about how we played. You can see how they celebrated when they won the game, we did that to them."