Postecoglou recalled Bissouma following a one-game suspension for posting a social media video that showed him inhaling nitrous oxide.
"You know what it's like when you punish your child and they do everything right for the next two days?" Postecoglou had said of Bissouma's return.
The Mali international made the most of his reprieve in the 14th minute, lashing home from the edge of the area for his first league goal for Tottenham in his 52nd appearance.
Son Heung-min doubled Tottenham's lead in the 25th minute, forcing Jordan Pickford into an embarrassing mistake as the Everton keeper was robbed by the South Korean for a simple finish.
Cristian Romero headed Tottenham's third in the 71st minute before Son bagged his second with a clinical finish six minutes later.
Bissouma marked his goal with a contrite celebration before running up to the touchline to hug his manager, who made it clear no more mistakes would be tolerated.
"Biss was good and we know Biss is a good footballer," Postecoglou said.
"It's about him being the best version of himself. Discipline is a big thing for him and he was disciplined today. It was a good performance.
"There's no doubting his ability to do something special. But he needs to make sure he does everything right on and off the field to give him the platform to show who he can be as a footballer. We certainly believe in him. It's up to him to keep going on."
Thanks to Bissouma's moment of magic, Tottenham were able to ease to their first win this season after a 1-1 draw in their opener at Leicester on Monday.
"The performance was excellent. We looked really threatening going forward all the time, created many different kinds of chances and handled their threat really well," Postecoglou said.
"We looked a threat from different areas. If we are going to turn compelling performances into consistent results that's what we have to do."
Successive defeats at the start of the season for the third straight year have underlined the size of the task facing Everton boss Sean Dyche.
In their last season at Goodison Park before moving into a new stadium, Everton are already facing another battle to avoid relegation.
"It's a strange situation here. We kind of build something, and then we have to go down the hill and build it again. What about playing when it's 0-0?" said Dyche, whose team lost 3-0 at home to Brighton in their opener.
"We've done well when the challenges have come our way. We've done great. So I reminded them of that afterwards. Reminded them that this is not our first rodeo.
"We get written off and then we start changing the storyline again. It's a never-ending challenge at Everton."
