Morgan Rogers had Villa ahead before Spurs overran the visitors through goals from Dominic Solanke (2), Brennan Johnson and James Maddison.
Postecoglou said afterwards: "Villa are a good side. The first half was a bit of an arm wrestle. They have a consistent level of performance. We have other gears in us which we showed in the second half.
"We stick to our principles, there is great belief in these players, and when we do that we are hard to stop. I felt the momentum was going our way, we knew we would finish strong and if we upped it a couple of gears like we can we knew we would be hard to stop."
On Heung-min Son being replaced, he explained: "He was never going to play more than 55 minutes. Last time he came back from injury he got to the 60-minute mark and got injured again. He got through unscathed."
On Solanke, he added: "Unbelievable effort. The work he did midweek against City. His goals are unbelievable but his general play, his work-rate, I can't speak highly enough of him.
"Seven days is a long time in football. Seven days ago I was a grumpy old so and so. We have to say true to the course we're on.
"We made sure not to feel sorry for ourselves and back to the team we want to be."