The move had been on the cards since last summer when new owners were brought in at Birmingham and rumours began to fly about Gold and Sullivan’s intentions for the struggling Premier League side.
"We’re taking on a huge task at a club with enormous problems and it will take time for us to turn it around," Sullivan told BBC Radio 5 live.
He and Gold will have complete financial control of West Ham and they have already assured that manager Gianfranco Zola will keep his job and that former Birmingham City chief executive Karren Brady will join as vice-chairman.
The pair are also looking for wealthy investors to help finance the club, which is £35m in debt, and are thought to have approached former rival bidders including Lotus F1 chief Tony Fernandes.
Sullivan added: "We have short-term goals and long-term goals. In the short term, we want to stay in the Premier League and in the long term we would like to be challenging for the top four and Champions League football.
"There will be some transfer funds available to the manager this month.
"January is not the best time to buy players – you don’t get many bargains in January – but the club has such an unbalanced squad, players will have to be bought, loaned or acquired.
"Myself and my partner David Gold will be honest with the fans about the books, the imbalance of the squad and the crazy wages the Icelandic owners paid out that has brought the club to its knees.
"But this is an important day for us. For 20 years, this is the club we wanted.”