A brace from Harry Kane and a third from Christian Eriksen was enough to see Spurs past their London rivals, despite a red card for Serge Aurier and two late goals from the Hammers, via Javier Hernandez and Cheikhou Kouyate.
While Spurs have won their first three games away of the season, for the first time in Premier League history, back at Wembley it has been a different story. A loss the Chelsea in their first game under the arch has been followed by disappointing draws against both Burnley and Swansea in the league.
Speaking after their first ever win at London Stadium, Pochettino described taking their away from to the national stadium as his "next challenge".
“I am so happy [with our away form],” Pochettino said. “If we see last season we have very good performance away from home.
“Now we need to start to feel good at Wembley, make home Wembley, try to win at Wembley I think that is our next challenge to win games there.
“I think the team is doing fantastic away from home, now we try to learn to improve like always and win games at Wembley."
Spurs won’t get their chance to rectify their home form in the Premier League until mid-October, with their next home fixture a clash with Bournemouth on October 14.