The result ensured West Ham climbed out of the relegation zone and Pellegrini, in turn, became the first manager to win a Premier League match on his birthday since Arsene Wenger's Arsenal beat Everton on 22 October 2017.
West Ham ended a 11-game run of conceding against Chelsea at home in the league when they won 1-0 in December - they last kept successive clean sheets at home versus the Blues in August 1969.
Wilshere's career has been punctuated by a series of injuries and boss Manuel Pellegrini revealed his latest problem has been tough for the player to accept.
Yarmolenko, who was signed from Borussia Dortmund in the close season, marked his first league start for West Ham against Everton with two goals, and Pellegrini said he was not surprised at the striker's quick adaptation to the English game.
West Ham will welcome Chelsea to the London Stadium this weekend in what usually turns out to be a feisty London Derby. More to follow. PLAYER RATINGS, LEAGUE TABLE AND MATCH ZONE Everton: Pickford, Kenny, Holgate, Zouma, Digne, Gueye, Schneiderlin, Walcott, Sigurdsson, Calvert-Lewin, Tosun. Now it is very hard because in the first part of the season it is very important to play with a very solid team.
Yarmolenko seems to get it. "He's having some treatment and we will see if he can be fit for Sunday".
Go for draws away and go for wins at home. So, why hammer a round peg in a square hole?
The backline and midfield three will be fine in their roles, but Anderson, Yarmolenko, and Arnautovic need the ball and need team possession to be effective. Patience will be key, just like at Everton, as West Ham will have to earn the ball off an impressive Chelsea team rather than just take it. However, the draw came in a moment when West Ham were extra motivated to stay in the Premier League (managed to stay with 42 points). "And when you get a performance like that as well, it gives you a lot of belief".