The hosts grabbed all three goals during an excellent first-half at Celtic Park with Greg Taylor, Daizen Maeda and Kyogo Furuhashi on target.
It continued the champions' superb form, while defeat for Livingston saw them drop to fifth.
Livingston manager David Martindale had no qualms about the loss.
"Celtic were worthy winners, to be honest. I thought they played some nice football at times, he said. "It was a really difficult game for us to lay a glove on them.
"They've got international players. I've seen them do that to Real Madrid - they were probably the better team in that game for 60 minutes.
"You've got to take your hat off to Ange. What he's done at Celtic, he's done it incredibly well.
"It's all right having money. It's how you spend it and then it's how you coach the players you bring in."
Postecoglou was pleased with his team's performance.
"I thought it was another strong performance. We knew it would be a tough challenge, trying to create opportunities in the game.
"I thought we were really good, we had a lot of possession but we used our possession really well, asking questions of the opposition, be threatening with our running.
"We scored three good goals, second half wasn't any easier because they still sat in and made it difficult."
Rangers kept the pressure on their Old Firm rivals with a fine 3-0 victory at Hearts.
Alfredo Morelos opened the scoring after nine minutes at Tynecastle before Malik Tillman added a second before the break.
Morelos wrapped up the points and completed his brace with 22 minutes left to keep up Michael Beale's strong start in charge at Rangers.
Aberdeen started life after Jim Goodwin with another defeat after St Mirren scored twice late on to claim a 3-1 win and move up to fourth.
Interim Aberdeen boss Barry Robson saw his tenure begin in the worst possible fashion with Ross McCrorie sent off with just seven minutes played for colliding with Charles Dunne.
Hosts Aberdeen managed to hold out until half-time before Curtis Main fired St Mirren ahead and, despite Bojan Miovski's 74th-minute leveller, Main grabbed his second with nine minutes left and Declan Gallagher hit a third on the stroke of full time.
Daniel Armstrong's goal helped Kilmarnock move off the bottom of the table with a 1-0 win over relegation rivals Dundee United.
Kilmarnock moved out of the bottom two with their victory after Motherwell lost 2-0 at home to St Johnstone following goals by Drey Wright and Melker Hallberg for the visiting Saints.
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