Previously goal-shy Everton tore through Spurs' under-strength backline to net three times before the break through Dominic Calvert-Lewin, Iliman Ndiaye and an Archie Gray own goal.
A first win in seven league games takes David Moyes' men four points clear of the bottom three.
Tottenham's 12th Premier League defeat of the season leaves them just four points above Everton in 15th and ramps up the pressure on manager Postecoglou.
However, it was chairman Daniel Levy, not the Australian, who felt the fury of the travelling support as they chanted for change at the top of the club.
"This is not about me, but what is about me is I have a responsibility to the players I do have. For me to focus on anything else is abstaining from the responsibility I do have," said Postecoglou, who was without 10 first-team regulars due to injury.
Everton did bite the bullet to sack Sean Dyche earlier this month and it was like the Goodison days of old for Moyes as he enjoyed the first win of his second spell in charge.
"Really pleased we could give everyone something to shout about," said Moyes.
"We played ever so well in the first half, scored three and could have scored more."
Moyes' homecoming fell flat in a 1-0 defeat to Aston Villa in midweek, after which the former Manchester United boss said he was aware of the task he faces to preserve Premier League status due to a lack of firepower.
Calvert-Lewin had not scored since September but did not look like a striker short on confidence as he turned Gray and Lucas Bergvall before firing in the opening goal on 13 minutes.
Spurs frailties
To compound the absence of first-choice goalkeeper Guglielmo Vicario and three of Postecoglou's preferred back four in Cristian Romero, Micky van de Ven and Destiny Udogie, Tottenham were also without club-record signing Dominic Solanke.
Postecoglou's men remain alive in three cup competitions, but Levy has a decision to make on whether the quest for a first trophy since 2008 would be better served by a change of manager.
"We're still in all four competitions. Our league position is not great to say the least but eventually our players will come back. Significant talent will come back," added Postecoglou.
"I've got great motivation to get through this so when we do get our players back we can get something significant out of our season."