After a subdued first 45 minutes, Spurs sprang into life early in the second-half as Joel Ward’s own goal and Son Heung-min’s strike earned the points.
Jordan Ayew pulled a goal back in stoppage time, but it was too little, too late for the Eagles.
Not since Spurs last won the league in 1960-61 have they enjoyed such an advantage at the summit of the English top flight.
A title challenge still appears a remote possibility, but with no European football to sap their energy, the sky may be the limit for Postecoglou’s men.
After Tottenham’s last game on Monday, the former Celtic boss was not fully satisfied with his side’s performance, even though they beat Fulham 2-0.
It was more of a mixed bag from the visitors as they took 45 minutes to find their rhythm.
Goalkeeper Guglielmo Vicario has been one of the unsung heroes of Spurs’ form and the Italian was needed to make a smart stop low to his left to prevent Odsonne Edouard opening the scoring inside the opening 10 minutes.
The signing of James Maddison has been the catalyst to Tottenham’s transformation under Postecoglou, despite the loss of the club’s record goalscorer Harry Kane to Bayern Munich.
Fired up by Postecoglou’s halftime team talk, Spurs emerged for the second period with greater intensity and Maddison created the opening goal as his driven low shot across goal was turned into his own net by the unfortunate Ward.
Postecoglou’s Midas touch was then in evidence as he introduced Brennan Johnson just after the hour mark.
The former Nottingham Forest winger was on the pitch just two minutes before he collected Maddison’s pass and crossed for Son to tap home his eighth goal of the season.
A fine night for Spurs had one final highlight as midfielder Rodrigo Bentancur made his comeback from an anterior cruciate ligament injury suffered back in February.
Ayew’s stunning strike into the top corner in the 94th minute was just Palace’s second goal in their last six games.
The Ghanaian escaped unpunished from a VAR check for handball to set up a grandstand finish.
Palace had one big chance with virtually the last kick of the game when Matheus Franca sliced wide, but Spurs showed their resolve to survive 12 minutes of stoppage time.