The Australian had already set an outright Premier League managerial record against Fulham on Monday night, where a 2-0 success saw him earn an unrivalled 23 points from his first nine matches at the helm.
Tottenham only had four days' worth of recuperation before meeting Roy Hodgson's Palace in another London derby at Selhurst Park, a ground where the Eagles have been notoriously tough to break down.
Postecoglou's side struggled to click into gear during a goalless first half, but they took the lead in fortuitous circumstances in the 53rd minute, as James Maddison's low cross cannoned off Joel Ward's leg into the back of the net.
Thirteen minutes later, the Premier League leaders doubled their advantage thanks to a silky combination between Maddison, Brennan Johnson and Son Heung-min down the left, which ended with the latter calmly tapping in from a few yards out.
However, the Eagles cut the deficit down to one through a pinpoint Jordan Ayew volley four minutes into injury time, with the goal eventually being given after a lengthy VAR check for handball.
Tottenham had their backs against the wall as Palace huffed and puffed for a leveller in the closing stages, but the North London giants held out for an eighth top-flight win of the season, opening up an unassailable lead over Manchester City and Arsenal in the process.
Not since their double-winning campaign of 1960-61 have Tottenham held such an advantage over the second-placed team, but as ever, Postecoglou refused to read too much into the fledgling rankings.
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