The Hammers are on cloud nine after wrapping up their campaign with a historic victory in the Europa Conference League against Fiorentina to earn their first major silverware since 1980.
While Moyes was the mastermind behind the success, it's been a less-than smooth season for the former Everton and Manchester United boss after a poor run of results saw his position come under threat.
Fans were calling for his head as West Ham drifted towards a relegation battle, however the club's hierarchy kept the faith and Moyes led them to a 14th place finish in the Premier League.
With a major European trophy secured as well, Moyes is now being lauded by the Hammers' fans, leading Sullivan to confirm their plans to keep the Scottish manager on board.
Speaking to Jim White, Sullivan told talkSPORT: "We believed in him.
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"We came in for a lot of pressure from supporters, the press and everyone who seemed mad on making changes but we believe in honouring contracts.
"David had a contract with us and has a contract with us until next summer and next summer we'll look at it again.
"Until then, if he wants to remain as manager then he will remain as manager of West Ham."
Moyes, who has been linked to the vacant Celtic job, hailed their Europa Conference League success as his greatest night.
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He was seen putting his winners medal around his 87-year-old father's neck.
"He was the first one, I thought I should get it round his neck," said the jubilant West Ham boss. "He's watched a lot over the years and I hope he enjoyed it."
Moyes added: "He's still going strong, still wants to come to the game. It's great, it was a great moment to have your family at the game.
"These sorts of moments as a manager don't come around too often. You have to work very hard.
"The family, from young boys, they always want to support you. You don't get enough good moments in football sometimes but today is a great moment for us."