The east London outfit secured a return to the top flight after beating Blackpool in Saturday’s Championship play-off final.
And, with the Hammers having made a bid to occupy the Olympic Stadium in the future, the former Birmingham owner predicts a bright future.
"West Ham are coming again. The days of decline are finished," he told The Daily Mirror.
"There is a very big gulf to jump but that is what we must aim for.
"And once we get all the debt paid off, which we hope we can do in three or four years, we can then build a team to take on the bigger clubs.
"It might be five or six years away but I don’t think the aim is to survive every year and scrap along at the bottom and pray you have a cup run."
Sullivan bought the club in January 2010 alongside business partner David Gold and inherited large debts from the previous owners.
However, he believes a period of consolidation in the top division will allow the club to compete with the top sides.
"If we can survive one year [in the Premier League] we can really start to build and move to the Olympic stadium," Sullivan added.
"I think the opposition is that we are not allowed to defend our corner at the moment because we cannot show people what we have got.
"I would say to the West Ham fans that if they like Wembley, they will like our new stadium because it will be very similar.
|5/4||West Ham are 5/4 with William Hill to be relegated in 2013.|
"The similarities are amazing and if we get the plan that we want it will be very special and our fans will be very proud of it and it will be an enormous legacy for East London.
"It is a wonderful stadium and if we are going to compete with Arsenal, Tottenham, Chelsea, Manchester United and Man City then we have to have a bigger stadium. And that is the aim."
The Hammers are favourites to move to the Olympic Stadium but have faced several obstacles in their fight to do so, having initially been granted the right to move to the ground only for the process to be blocked by legal challenges from Tottenham and Leyton Orient.
The bidding process is due to conclude in October, with the stadium scheduled to re-open in the summer of 2014.