Celtic stayed in sixth place in the cinch Premiership, six points behind the leaders, having collected 10 from seven matches.
The hosts hit the bar through David Turnbull, Tom Rogic and Jota and Albian Ajeti somehow blazed over from two yards.
But United also had chances to win it after Ian Harkes quickly cancelled out Liel Abada’s early headed goal with a similar effort.
Visiting boss Tam Courts was adamant United should have had a first-half penalty when Cameron Carter-Vickers barged over Nicky Clark only for referee Kevin Clancy to play on.
Joe Hart also made a stoppage-time save from United substitute Louis Appere as Celtic saw their injury concerns mount further with Josip Juranovic and James McCarthy both going off.
Postecoglou already had eight players missing and praised his team’s effort.
And he claimed taking any wider implications for their season was premature.
“It’s a weird league,” he said. “You call things early here, don’t you? It’s quite remarkable that seven games in people are calling the title already. It’s just not how I work.
“I’m not pulling up stumps after seven games just because other people seem to think there is some sort of insurmountable challenge out there for us.
“If we weren’t playing well, if we were struggling through things, then by all means I would be looking at things a lot differently.
“With the efforts we are putting in at the moment, and the football we are playing, and the chances we are creating, and what we have got to come back into the team, I just see it totally differently.”
When reminded about the demands at Celtic, he said: “I know people are wanting to push me on some kind of edge of the ledge somewhere. It’s just not going to happen.
“I don’t worry about those things. I look at how my team is playing and what we are trying to create.
“I know what the club is about, I know what the expectations are.
“I believe the road we have started on will get us there so we will just keep going.”
Postecoglou felt his side’s efforts warranted three points.
“The players are disappointed but as I said to them, the result is on me,” he added. “The performance level they put in with three games in a week without changing the team too much, not just this week but right through this period, right to the end there they were still giving everything and still creating chances.
“I couldn’t praise them high enough for the efforts they put in.”
United boss Tam Courts also felt his side deserved better.
“First and foremost, I am really proud of the players for their effort levels and quality and even their bravery to come here and put on a performance like that,” he said.
“But I am also slightly frustrated because in the first half we were denied a stonewall penalty and then we had a chance right at the end there to probably take all three points.
“I accept that Celtic had a couple of chances and hit the woodwork, but I thought we probably just merited the three points on the balance of play.”
Having been exasperated by several decisions during a midweek cup defeat by Hibernian, Courts did not speak to Clancy about the penalty claim.
“I am standing diagonally behind it and I can see there is a barge on the back,” he said. “It is a huge call at a huge stadium with a lot of fans here. Also, the referee might have had to produce a red card.
“So I think, for whatever reason, he has decided not to award the penalty. So hugely frustrated that in the space of a couple of days we have been at the mercy of a few bad decisions.
“But on this occasion I will show a bit of professional decorum. But I am really expecting better from the officials.”