Everton manager Carlo Ancelotti told BBC Sport: "The game was in the balance. We didn't have a lot of opportunities; they didn't have a lot of opportunities.
"We have to accept the result. It will be important to keep our belief high.
"We are more comfortable defending and using the counter-attack. When we have the ball we have more difficulties and we have to adjust this.
"We are in the position we would like to be and hopefully we can be there at the end of the season. That is our target."
West Ham manager David Moyes said winning at Everton for the first time since he left the club was "special".
Tomas Soucek's late goal gave West Ham a 1-0 Premier League victory at Goodison Park.
"Always going back to your old clubs is special," said Moyes, who had lost on his previous four visits to Everton following his departure in 2013 after 11 years in charge.
"I had a brilliant time here - great years - but I want to do it again at West Ham and that means going and winning at the other clubs."