Everton kicked off their campaign with a lacklustre 2-2 draw against promoted side Watford but have only lost to Manchester City and have scored eight goals while keeping three clean sheets.
The Merseyside club, yet to lose an away fixture this season, travel to The Hawthorns to face West Bromwich Albion on Monday, knowing a win could see them climb to fifth in the table.
"We are definitely satisfied with how it's gone and the bigger picture is that we've made a real solid start to the season," England international Barry told the Liverpool Echo.
"After a draw at home on the first day of the season, where we were disappointed, and then losing to Man City is no disgrace, we have picked it up, kept some clean sheets and have given ourselves something to build on."
Barry's midfield partner James McCarthy echoed the comments of his team mate but acknowledged it was early days.
"Of course we want to build on the start we've had and we believe we can. We want to achieve things at the club, get into European places, that's the aim," the 24-year-old McCarthy told the club's website (www.evertonfc.com).
"But we won't look too far ahead because it is only the start of the season.
"Come January we can look again at the table and take stock then. There will still be a long way to go in the league but we can weigh up from there where we want to be going."
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Ian Ransom)
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