Having scored just once in his first 45 Premier League appearances for Tottenham, Emerson Royal has since netted two in his last five
The Spurs wing-backs linked up for the first goal of the game, with Ben Davies releasing Emerson Royal to shoot into the bottom corner.
West Ham had an immediate chance to equalise but Jarrod Bowen's powerful effort, after he outpaced Cristian Romero, was well saved by home goalkeeper Fraser Forster.
Spurs substitute Son Heung-min scored four minutes after coming on when he raced onto Harry Kane's through ball and slotted a finish past Lukasz Fabianski.
The Hammers would have moved above both Bournemouth and Everton with a draw, but instead remain 18th after a first defeat in five matches in all competitions.
"We've only got ourselves to blame for it, and overall the players did a lot of good things as well," Moyes told Match of the Day.
"We've been here before so I'm sure we can do it again, but I'm looking forward to the players who are joining me in the fight because that is going to be noticeable in the weeks to come. They're all for it."
Speaking to Sky Sports, he added: "There are bits of it I want to build on and hope that we take it in. The players have been challenged, now let's see who's up for the fight."
Tottenham, meanwhile, moved one point ahead of Newcastle, although Eddie Howe's side, beaten 2-0 at home by Liverpool on Saturday, have a game in hand.
Tottenham had assistant manager Cristian Stellini back in charge with Antonio Conte resting after his recent gallbladder surgery.
Conte was on the touchline for the recent games against Leicester and AC Milan, but subsequently said he had rushed his recovery and was not ready for a full-time return.
Harry Kane is only the third player to reach 200 Premier League goals
In Tottenham's last home game, the 1-0 win over reigning champions Manchester City on 5 February, Kane scored his 267th Spurs goal to take him above Jimmy Greaves' total and become the club's all-time record goalscorer.
Before this match there was a celebration of that achievement as Kane was handed a silver boot trophy, with members of the Greaves family making the presentation.
Tottenham's fans also spelled out the word 'Harry' with an impressive Tifo before the game, with Kane's children, wearing Spurs shirts that said 'Daddy 267' on the back, joining the England captain on the pitch.
However, he could not find his 268th Tottenham goal, shooting wastefully wide with only Fabianski to beat midway through the second half from his only notable chance.
But Kane did manage to record an assist, with his pass through to Son shortly after the South Korean, dropped for Richarlison, had come on as a substitute.
Towards the end of the match, the home fans taunted the away supporters with chants of "West Ham are going down" and it was a disappointing performance from Moyes' side after they had gained decent draws against Newcastle and Chelsea in their previous two Premier League games.
The visitors had wasted a good chance to take the lead inside the opening minute but England forward Bowen sent a 20-yard half-volley just wide.
Bowen had a chance to make it 1-1 but was denied by Forster with what proved to be West Ham's only shot on target.
Tottenham are still in this season's Champions League but lost 1-0 away to AC Milan in Tuesday's last-16 first-leg tie and this result puts them in one of the four automatic qualifying spots for next season's tournament.
In the end, Spurs ran out deserved winners against West Ham as a controversial video assistant referee call did not come back to prove costly.
The hosts thought they should have earlier been given a 12th-minute penalty when Thilo Kehrer, standing close behind Richarlison and falling over, put his hand down and handled the ball.
But a decision was not given by on-field official Michael Oliver, with those in the VAR room not telling him to review the decision.
-- Courtesy of BBC Sport