The Hammers opened the scoring in the 18th minute when Jarrod Bowen fired a ball across the box from the right of the six-yard box towards an unmarked Mohammed Kudus, who swept past Guglielmo Vicario.
Dejan Kulusevski equalised for the hosts in the 36th minute after cutting inside from the right of the box and seeing his strike bounce off both posts before trickling in, before Yves Bissouma gave Spurs the lead in 53rd minute when Destiny Udogie teed the midfielder from inside the penalty area.
Postecoglou's side took a 3-1 lead just two minutes later courtesy of an own goal from Alphonse Areola after the ball deflected off Jean-Clair Todibo and onto the goalkeeper following a shot from Son Heung-min, but the forward got his goal in the 60th minute after carrying the ball into the penalty area before beating Todibo and Areola.
The victory means Spurs have now collected 13 points from eight matchweeks and they are now just one point behind fourth-placed Chelsea.
As for West Ham, they have won just three times in 10 games under the management of Lopetegui, and are in 14th place with only eight points from eight Premier League games, just five points more than 18th-placed Crystal Palace.
After his strong showing against Ipswich Town last time out on October 5, Bowen continued his fine form and registered his second assist of the season after he twisted past Udogie and played towards Kudus, who had been left free in the box by Pedro Porro.
Kulusevski levelled the scoreline when James Maddison carried the ball forward during a counter-attack before passing to the Swede who cut inside from the right and struck a shot off of the right post, which bounced onto the left and eventually crossed the line.
Spurs blitzed the Hammers in the second half, with Bissouma scoring after Udogie made an underlapping run into the penalty area and cut back towards the edge of the box, before Son forced an own goal when his effort was deflected onto Todibo and then Areola.
Pape Sarr brilliantly won the ball back from inside his own half, playing a defence-splitting pass into West Ham's half to Son, who carried the ball into the box and beat Todibo with trickery and beat Areola with a left-footed from the left of the six-yard box into the left of the net.
TOTTENHAM HOTSPUR VS. WEST HAM UNITED HIGHLIGHTS
Mohammed Kudus goal vs. Tottenham Hotspur (18th min, Tottenham Hotspur 0-1 West Ham United)
Dejan Kulusevski goal vs. West Ham United (36th min, Tottenham Hotspur 1-1 West Ham United)
Yves Bissouma goal vs. West Ham United (53rd min, Tottenham Hotspur 2-1 West Ham United)
Alphonse Areola own goal vs. West Ham United (55th min, Tottenham Hotspur 3-1 West Ham United)
Son Heung-min goal vs. West Ham United (60th min, Tottenham Hotspur 4-1 West Ham United)
MAN OF THE MATCH - DEJAN KULUSEVSKI
Prior to Postecoglou's arrival at Spurs, Kulusevski was seen by many as a creative winger but the Swede has been allowed to showcase his creative abilities in more central zones, including against West Ham on Saturday afternoon.
The Spurs man produced one goal, created two chances and played four passes into the final third, and managed to excellently link play from midfield to the forward line.
On top of his strong showing on the ball, Kulusevski was also combative out of possession, winning eight of his 14 duels and making eight recoveries.
TOTTENHAM HOTSPUR VS. WEST HAM UNITED MATCH STATS
Possession: Tottenham Hotspur 57%-43% West Ham United
Shots: Tottenham Hotspur 22-11 West Ham United
Shots on target: Tottenham Hotspur 7-4 West Ham United
Corners: Tottenham Hotspur 13-5 West Ham United
Fouls: Tottenham Hotspur 10-15 West Ham United
BEST STATS
WHAT NEXT?
Tottenham are set to host AZ Alkmaar on Thursday in their third Europa League game of the season, before travelling to Selhurst Park to take on Crystal Palace on Sunday.
Meanwhile, West Ham have no midweek game but will play Manchester United on Sunday at the London Stadium, before facing Nottingham Forest on November 2.