Burnley failed to record a win in their first 10 top-flight matches following their promotion from the Championship last season, but Barnes' header at Turf Moor secured a long-awaited victory.
The striker insists the players never stopped believing they could compete in the Premier League, and Saturday's win has only strengthened that opinion.
"We believe and our changing room believes," he is quoted as saying by the Lancashire Telegraph. "A lot of people don't believe and maybe have written us off already but we believe and that's all that matters. We believe in our own ability.
"We are now going into the international break on a high and then we come back, work hard and go again.
"We will take it one game at a time and see where we go. But like I say, we believe. We are going out to win. If we keep playing like that there's every chance we can stay in this league."
Barnes' winner against Steve Bruce's side was his first goal in the the top flight and the former Brighton and Hove Albion man hopes there will be plenty more to come.
"I have played in nearly every league possible, right from the bottom," he added. "So it's a dream come true to play in the Premier League and to get off the mark and it's the first of many hopefully.
"I just threw my head at it and it paid off. Maybe our luck is changing which is what we need."
