Spurs were held 1-1 by Leicester City in their opening match but showed no such signs of profligacy against Everton, with Bissouma setting the tone inside 14 minutes with a 20-yard rocket, his first goal for the club.
Jordan Pickford's howler 11 minutes later after dallying too long on the ball gifted Son a simple finish at Tottenham Hotspur Stadium as Everton, beaten 3-0 by Brighton in its opening match last week, caved under the pressure.
Everton offered a little more attacking impetus in the second period with the arrival of recent signings Dame Ndiaye and Jesper Lindstrom, the latter forcing Guglielmo Vicario into a rare save.
However, Cristian Romero's towering header on his 100th Tottenham appearance put the hosts out of reach on 71 minutes, followed soon after by Son's second after a brilliant run from Micky van de Ven.
Everton remains planted to the bottom of the early standings as one of four sides with zero points after two matches, while Tottenham has four points from its two games.
Everton has now lost their its two games by an aggregate 7-0 scoreline, with this marking the first occasion it has lost its first two games in a top-flight season by three-plus goals in its history.
It is a familiar losing feeling for the Toffees, as it becomes just the second side to taste defeat in its opening two Premier League games in three successive campaigns after Bournemouth between 2015-2016 and 2017-2018.
But it was happy days for Postecoglou, the Australian seeing the attacking prowess of his side on full display.
But he said Bissouma "still has work to do" to win back his full trust, although marking his Tottenham return with a goal will certainly help.
Bissouma was suspended by Spurs after filming himself inhaling nitrous oxide and missed the 1-1 draw with Leicester City as a result.
Asked about Bissouma after the win, Postecoglou said: "It's a good start. He's still got work to do.
"I'm not doubting his footballing quality. We want to make him the best version of himself. That starts off the field as much as on it."
After beating Everton by four-or-more goals for the fourth time in the Premier League, Postecoglou insisted there is still more to come from his side.
"The performance was good. It's really encouraging for us," he said. "There is more [to come]. We want to keep performing."
Asked if he will add to his squad before next week's transfer deadline, Postecoglou added: "I think we have spent wisely.