Tamworth took the tie into extra-time before Spurs won 3-0 via an own goal from Nathan Tshikuna before the visitors struck through Dejan Kulusevski and Brennan Johnson.
"It was a challenging game as we expected it to be," said Postecoglou afterwards.
"They made it hard for us, with the surface it was about us staying clam and being persistent in our football.
"In the end, we got the job done."
On his message going into extra time, he continued: "Stay calm and persistent. We knew we would finish strong and you could see they were tiring a bit.
"It's a game of football you go into it with trepidation and knowing you can slip up with the challenges.
"Most parts the boys kept their heads. I thought they stayed calm and persisted and got through.
"We are through to the next round and that was the objective for today.
"Tamworth did really well and should be proud of their efforts. For us, we move on."