The Tigers have enjoyed a positive start to the campaign, with three wins from seven matches lifting them up to eighth place in the table.
Hull kept their third consecutive clean sheet last weekend when they drew 0-0 with Aston Villa at home.
"I expect us to stay up. I expect us to win every game if we can," Rosenior told the Hull Daily Mail.
"The Premier League has learnt that we're going to be a hard team to beat. We don't lie down for anyone and we give our all every game.
"You won't see too many teams beating us by four or five this season, that's for sure."
Hull's next fixture in the Premier League is a trip to Everton on October 19.