However, regardless of the scoreline, Spurs were dominant for the most parts of this London derby as Mourinho witnessed his new players put in a much-improved performance after the departure of Mauricio Pochettino.
While Spurs move up from 14th position to sixth spot in the Premier League standings, West Ham remain in 16th spot after a fifth defeat in six top-flight outings - a run which leaves Manuel Pellegrini fighting to keep his job.
Mourinho took the opportunity to move Eric Dier into midfield and hand a recall to Moura, and the pair both played their part as Spurs dominated the early stages in East London.
Kane had the ball in the net as early as the fourth minute, but the assistant correctly ruled that the forward had been in an offside position before finding the roof of the net from 10 yards.
While West Ham responded with Sebastien Haller having a header deflected into the grateful hands of Paulo Gazzaniga, Spurs largely remained in the ascendancy with Dele Alli particularly impressive in the final third.
Son had a long-range effort tipped away from goal by Sergi Roberto, and the visitors maintained their threat going forward before finally taking the lead nine minutes before the break.
Alli was again instrumental as he picked out the run of Son, who fired a low shot into the far bottom corner, although Roberto was at fault for failing to get a stronger pair of hands behind the ball.
West Ham needed the half-time whistle with Alli twice threatening with efforts from distance within the space of five minutes, but Spurs were able to double their advantage shortly afterwards.
After brilliantly keeping the ball in play on the touchline, Alli released Son down the left, and after firing a cross towards the far post, Moura was able to get ahead of Aaron Cresswell to convert from six yards out.
West Ham were fortunate to avoid falling further behind in the third minute of added-on time as Davinson Sanchez saw a header tipped over the crossbar by Roberto, with the half-time whistle consequently being greeted by a chorus of boos.
Moments after the restart, Moura should have added a third for Spurs, with the Brazilian dragging a shot from 10 yards well wide of the post after being played in by Son.
However, a minute later, the visitors extended their lead through Kane, who got onto the end of a right-wing cross from Serge Aurier to power a header past Roberto from 10 yards.
That goal knocked the stuffing out of West Ham and although substitute Antonio tested Gazzaniga with a rising shot from distance, there was a feeling that Spurs would be able to coast through to the final whistle.
Antonio, who had impressed since his introduction, had other ideas, however, with the winger finding some space on the edge of the area to drill a low effort into the bottom corner of Gazzaniga's net.
West Ham had the momentum heading into the closing stages but chances for the home team remained few and far between, with a Declan Rice tap-in which was correctly ruled out for offside by VAR representing their solitary clear-cut opening.
While the Hammers scored a second consolation effort with what was the lack kick of the game as Ogbonna netted a volley from a corner, Spurs were never in any real danger of failing to register their first away win in the Premier League since January.
WEST HAM UNITED (4-3-3): Roberto; Fredericks, Ogbonna, Diop (C.Sanchez 64'), Cresswell; Snodgrass, Rice, Noble; Anderson (Antonio 46'), Yarmolenko (Fornals 56'), Haller
TOTTENHAM HOTSPUR (4-2-3-1): Gazzaniga; Aurier, Sanchez, Alderweireld, Davies (Rose 75'); Dier, Winks; Moura (Sissoko 82'), Alli (Eriksen 79'), Son; Kane