The 25-year-old England international struggled to have an impact at Upton Park in 2013-14, scoring just two Premier League goals in 15 appearances after a lengthy injury layoff.
Having paid Liverpool about £15 million to sign Carroll permanently, more will be expected of the former Newcastle man.
Sullivan was pleased Roy Hodgson left Carroll out of his England squad for the FIFA World Cup and hopes to see more from the striker in 2014-15.
"I still think Andy Carroll will come good. He didn't have a pre-season last year. He never really got his act together because of that and I think you'll see a much better player next season," Sullivan told The Mirror.
"It was a fair decision [leaving Carroll out] based on what he did last season. He didn't do enough when he came back.
"There were a few games he was devastating in - against Swansea he set up both of Kevin Nolan's goals - but he knows he didn't do enough to play for England."
He added: "I'm pleased he wasn't with England this summer. His time will come."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.