With a three-point gap to make up to Aston Villa in the Premier League standings, Tottenham would have momentarily risen into the top four of the table with victory by a two-goal margin or more at the London Stadium.
The Lilywhites made the short trip to the Hammers' headquarters on the back of a nervy 2-1 success over Luton Town, where they failed to score in the first half of a Premier League game for the sixth time in a row.
However, Tottenham exorcised those demons just five minutes into the midweek contest, as Timo Werner's crisp delivery found Brennan Johnson - promoted back to the starting lineup in place of Dejan Kulusevski - to tap home.
Postecoglou's men only held their advantage for 14 minutes, though, as their set-piece frailties reared their ugly heads once more when Kurt Zouma rose highest from a corner and equalised via an unorthodox effort off his back.
Tottenham drop more points from winning positions
David Moyes's men also let a couple of promising chances to take all three points pass them by, as Lucas Paqueta fired wide from the edge of the box while Guglielmo Vicario came out on top in a one-on-one with Michail Antonio, although Paqueta appeared to foul James Maddison before the latter chance.
Nevertheless, by letting a one-goal lead slip at the London Stadium, Tottenham have now dropped 20 points from winning positions in the current Premier League season, albeit while also gaining 22 from being behind
When asked in his post-game press conference why Tottenham were missing a "clarity of thought" on Tuesday, Postecoglou responded: "Because they are human beings, it is just football," as quoted by football.london.
"I would love to have a joystick and put them where I want them, but it does not work like that. Our role is to try and help them through that. It is not easy, again when you're facing a team like West Ham, physically they are a big side and they sit deep so it feels constricting anyway.
"I just think at time we have more time than we think. We scored a great goal but there were others times when the ball was flashed across the box when we should have been in those areas. Those are the things we're working on but hopefully we can help our guys in that front third."
Postecoglou plays down James Maddison concerns
For the third game running, Tottenham boss Postecoglou hauled off creator-in-chief James Maddison midway through the second half, having also withdrawn him before the 70-minute mark in clashes with Luton and Fulham.
In his nine top-flight appearances since recovering from a severe ankle injury, Maddison has only completed the full 90 minutes on one occasion, but Postecoglou has stressed that there is nothing wrong with the 27-year-old physically.
"He's not carrying a knock. If he was carrying a knock he wouldn't be playing. He's fine, yeah," Postecoglou said, before going on to praise Maddison's display in the London derby draw.
"I thought Madders was really good, especially when he got in their final third. I thought he played some really clever passes that we just didn't capitalise on. I think for all the guys in the front third area - not just the attacking players, the full-backs as well, we got into some really good areas, it's just about trying to show them that they don't need to be so rushed in their decision making and we'll get better outcomes."
Tottenham now have four full days to rehabilitate before welcoming Nottingham Forest to North London on Sunday, where Tricky Trees boss Nuno Espirito Santo will reunite with his former employers.