The Hammers ended their dismal run with a deserved 3-1 win at Everton to move out of the bottom three with only their third victory over the Toffees in 21 meetings.
“In football you are always under pressure,” said the Chilean, who was celebrating his 65th birthday.
“Because we won today we are not going to have pressure. For me, the pressure is the same in every game, the pressure to win.
“I have a lot of communication with the owners, of course they were not happy but they never have any doubt about what we are doing.
“I am very happy with the performance: first because this is a very difficult stadium to win, second because the players didn’t have any doubts.”
Pellegrini was rewarded for handing £17.5million summer signing Andriy Yarmolenko his full Premier League debut with two goals from the Ukraine international to put the Hammers 2-0 up just past the half hour.
Gylfi Sigurdsson’s header pulled one back a minute before the interval but Marko Arnautovic’s second half strike confirmed Everton’s first defeat of the season.
“He didn’t arrive here ready to play,” added Pellegrini.
“He has a lot of quality and he always demonstrated it but we preferred to work with him for a couple of weeks.”
The Hammers boss also played down suggestions substitute Lucas Perez had refused to play when Arnautovic went down with a knee injury just past the hour.
Pellegrini said Perez had finished his warm-up and had sat back down while Michail Antonio was just completing his and was ready to go on immediately.
“It is very simple. When Marco has a problem with his knee I want to put on Lucas but Michail (Antonio) was ready.”
Everton manager Marco Silva was left to rue costly errors.
“We didn’t perform in our level to win this match. I already told our players,” he said.
“In this level you cannot make mistakes like we made. When you create chances you have to score.
“After the first mistake we made we didn’t react like we had to react and it looks like the game changed completely and we lost our confidence.”
At 2-0 down Silva substituted Morgan Schneiderlin, whose father died in the week, a minute before half-time but the Toffees boss insisted his decision was a tactical one.
“I know what has happened but he was ready to play,” he added.
“It is not because of what happened or because of the performance of Morgan. He told me he was ready to play and he has all my support.”
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