"We weren't good enough to win the match, that's clear to me", he said.
"On some days maybe with West Ham 2-0 up they could come back but we must be able to repeat our performances".
West Ham will look to get their first points on the board when they travel to Merseyside today to take on Everton at Goodison Park.
Chilean boss Pellegrini, celebrating his 65th birthday on Sunday, had spent £100 million ($129 million) bringing nine new signings to the London Stadium in the close season.More news: Hurricane Florence claims first victims as tree kills mum and baby
It was a Pellegrini signing who opened West Ham's account, with winger Andriy Yarmolenko scoring his first goal on his first Premier League start.
"His old Stoke [City] team-mate Xherdan Shaqiri can afford to have 20 out of 21 erratic games if he pitches up in that last one and scores two worldies because Liverpool aren't relying on him, he's not the man".
"I am anxious about finding a team to play well and win the game".
Writing in his column for the Times, Souness said: "We don't know for sure who has the final say on signings".More news: Reasons to buy Apple's iPhone XR instead of an iPhone XS
Arnautovic has worn the armband in the absence of Mark Noble in recent weeks, and both of West Ham's Premier League goals this season have come from the forward.
But this result eased the pressure and, despite the optimism surrounding Silva's arrival in the summer, puts West Ham just three points behind Everton, despite starting with four successive defeats. Me and Marko (Arnautovic) were in the thick of that in the changing room and sometimes that's what you need. I was anxious about the game, because I was absolutely convinced we must end these bad results.
The coach is new and you have a lot of new players.
"The players talk to other players but I'm not anxious about that, I'm anxious more about trying to win three points".More news: Ian Wright's ideal response after Alan Shearer claims Arsenal aren't that good
As for Everton, Silva will not have long to rectify the problems he saw on the pitch.