The hosts fell behind after just 23 seconds at the Tottenham Hotspur Stadium, as Bryan Mbeumo volleyed Thomas Frank's side into an early lead.
Yet, Spurs responded well and equalised through Dominic Solanke, while Brennan Johnson completed the turnaround inside the opening half an hour.
Postecoglou's side registered 23 shots on goal to their opponents' six during an impressive attacking display, but it was not until the 85th minute that they sealed the win through James Maddison.
"It was a good win, a good performance," the Spurs boss told BBC Match of the Day. "Our performances have been good all year, but today, we got the result as well, which is the most pleasing.
"We should have won by a fair bit more, but we still got the job done. You're always keeping the opposition in the game [at 2-1], and it's been the story of our season so far. It was important to finish the game off."
Postecoglou also paid tribute to Solanke, who scored his first Tottenham goal since arriving from Bournemouth for a club-record fee of £65million.
"Dom worked his socks off. He was gone at the end," the Australian said. "He is still getting back to match fitness, but I know he is going to provide so much for us in that central position.
"All strikers love goals, so it will be a great moment for him, especially at home."
"[It feels] fantastic," Solanke added. "As a striker, it's always important to get that first goal and I managed to get that today, which I'm happy with. Hopefully, I can continue in this way.
"We have been playing well all season, but sometimes, you don't get the results you deserve. But if you keep going, keep sticking at it and playing the way we want to play, it's going to happen.
"To get the win here meant a lot to all of us. We have had some great performances to start the season, but haven't managed to get the result to match that. Today, we did that."
