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.
"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."
