Arteta took time to build a side that could compete with the best in the country, and during his early struggles, there were plenty of doubts about why he could not bring quick success.
Postecoglou's methods have been questioned recently by the more sceptical Spurs supporters, especially after Tottenham's costly capitulation at Brighton in their previous outing a fortnight ago, when a 2-0 lead became a 3-2 defeat after a disastrous period at the start of the second half.
Postecoglou explained on Friday that performance data backed up what his eyes told him – the players dropped their running levels by 20 to 30 per cent after half-time, and their intensity drop-off allowed the opposition to seize control.
So he was especially pleased that Tottenham's players had learned their lesson, and stepped up their intensity after the break to see off a stubborn but limited West Ham side who had gone ahead through an early Mohammed Kudus goal.
With the mood music around the club far from harmonious following the Brighton defeat, that setback could have set alarm bells ringing among supporters, but they and their team were instead spurred on, Dejan Kulusevski equalised before the break.
After half-time, Tottenham increased their intensity and three goals in eight minutes secured victory.
“That's part of the evolution of the team and where we want to get to,” said Postecoglou.
“I made it pretty clear after the Brighton game, there's certain principles underpinning our football which allows us to be the expansive and creative team we are. There's some real emphasis on hard work, organisation and discipline and we had to show both today.
“I don't think we would've been as dominant in the second half if we'd not had that underpinning of a strong foundation there. They're a big, physical team and you have to match them in certain areas.I t was a bit of an arm wrestle at the start but I thought we maintained our intensity and eventually broke them.”
Yves Bissouma put Spurs ahead in the 52nd minute, the returning Heung Min Son forced an own goal from Alphonse Areola, and then the Korean made it 4-1 on the hour mark.
“Second half we went up a gear and again I thought our football was excellent,” added Postecoglou.
“Part of the momentum we built came through the edge the supporters gave us, especially after the second goal; you could really feel the atmosphere inside the stadium lift and that pushed us along.
“We had a 15 to 20 minute spell where we were outstanding. It goes hand in hand. We need the supporters especially at home to provide a lift, but we need to give them something in return. Today is hopefully a little bit of a reward for our fans.”
Postecoglou showed good in-game management, too, showing the flexibility needed to take Tottenham to victory when he replaced playmaker James Maddison with the more physical Pape Matar Sarr at half-time.
“West Ham ask you certain questions in midfield and I just felt Pape's running power would help us in the second half, give us some real energy. I thought he did really well and he gave us a platform to really threatening every time we want forward and clinical in our football.”
Spurs could and should have had more as West Ham showed the sort of collapse Spurs had suffered two weeks earlier. Son hit the post, his replacement Timo Werner missed two chances, and Dominic Solanke went close after a mazy dribble from his own half.
The Hammers have only won twice under Julen Lopategui, against strugglers Ipswich and Crystal Palace, and have conceded 20 goals in their ten games in all competitions.
Questions are being asked about the former Real Madrid and – briefly – Spain manager, who admits his communication with his players needs improving, particularly with Kudus, who is set for a three-match ban after being sent off in the closing stages for pushing Micky Van de Ven in the face.
The Ghana international was again West Ham's best player, prompting two good saves from Guglielmo Vicario as well as scoring, but his relationship with Lopetegui is said to be strained.
“I am not going to say here what I am going to talk to him about, but for sure we are going to talk,” said the manager.
“We are going to have time and I am sure he will be aware that he can improve his answer in this moment because it’s not good for him, for the club and not good for the team.”
Kudus will miss the visit of Manchester United next weekend, a game West Ham need to win if they are to avoid getting drawn into a relegation battle, sitting in 15th place just five points above the drop zone.
Spurs, meanwhile, are back on track for a tilt at a Champions League place, just four points off the top four and with games against Palace and Ipswich to come in the league. Having learned a harsh lesson at Brighton, now could be the time for Tottenham to kick on – and for their supporters to trust Postecoglou's process.
TOTTENHAM: Vicario 7; Porro 7, Romero 7, Van de Ven 7, Udogie 6 (Gray 87); Kulusevski 8, Bissouma 8 (Bentancur 80), Maddison 7 (Sarr 46); Johnson 6, Solanke 8 (Richarlison 80), Son 7 (Werner 70).
WEST HAM: Areola 7; Wan-Bissaka 6, Todibo 5, Kilman 6, Emerson 6; Paqueta 6 (Alvarez 65), Soucek 6 (Soler 65), Rodriguez 5 (Mavropanos 70); Bowen 6, Antonio 5 (Summerville 65), Kudus 6.
Referee: Andrew Madley 8/10.