The Baggies finished 8th last term, recording nine home victories - the highest in a single Premier League season.
And Brunt is confident Steve Clarke's men can build on their success, starting with the visit of Southampton on the opening day.
"Expectation levels rise every year," said the Northern Ireland international.
"Realistically we did as well as we were going to do last season.
"It's our job as players to try and kick-on and do a little bit better this season. We can build from there.
"The fans will expect it and when you start being a little bit more successful you have to deal with that and I think we've got a good group of players who are more than capable of dealing with that."