"We need to show more quality in order to stay up", said Frank Lampard, "but we are confident that we will stay up."
The Toffees are 17th in the table and are only three points above the bottom of the table.
We have 10 games and a lot of points to fight for, but we are in this position and know the reality.
I have a strong belief that we will be okay if we keep on the path.
The first half ended with West Ham going ahead through a superb first-half free-kick by Aaron Cresswell.
The Toffees, who have six away points, fell behind again five minutes later to the winning goal.
After his side lost for the sixth time in eight league matches since he replaced the sacked Rafael Benitez in January, FrankLampard said "We have to show more quality - and there is that in this group - to stay up."
Everton has a large fan base. We are working to get through this moment. The players and myself are giving everything. The fans can know if we keep doing that.
The Hammers boss expects his old club to maintain their record of playing in the top flight every year since 1954.
When they have the quality of players they have with a Brazil centre-forward and an England centre-forward, I think they will be okay.
Everton have good things going for them and should feel more positive. It is not a good place to be and you never know how it will go, but the quality of the other clubs is not as good.
Jonathan Woodgate felt the players needed to work harder to ensure their safety.
The five forwards that are forward don't get back in time when they lose the ball, according to Woodgate.
If you look at how easy they are to play through, there is no real desire to make a difference.
You cannot always look at the managers. Sometimes the buck has to stop with the players.
A broken foot ended the hopes of an England call-up for the friendly against Switzerland and the game against the Elephants.
The 25-year-old forward, who has now scored 13 goals in all competitions this season, could make a late push to get into the England squad for four Nations League games in June.
It was unfortunate that he got injured, as we could have found him in the England squad.
If he keeps his form up, gareth may look at him again in the summer. We know that he has goals.
He might have been called up for England in the last break because of his high standards. He will be in contention for the games in June if he finishes the season well.
When they take on Lyon in the first leg of the Europa League quarter-finals at London Stadium on Thursday, David Moyes will be thinking about Barcelona.
The Hammers boss added that the main course was coming up on Thursday.
West Ham have never played in the top flight of European football, but if they win the Europa League, they will get a place in next year's competition.
Their win on Sunday leaves them sixth in the league with 51 points, three behind fourth-place Spurs. The Hammers have three games in hand, while Spurs have a game in hand.