A statement on West Ham's website confirming Moyes's departure said the club wanted to take a "different direction".
Moyes' assistants Alan Irvine, Stuart Pearce, and Billy McKinlay have also departed from the London club along with the 55-year-old.
Moyes took charge of West Ham in November a year ago, signing a six-month deal following Slaven Bilic's sacking, with the club in the relegation zone.
West Ham confirmed Moyes's exit on Wednesday and Sullivan thanked him for his efforts after becoming a member of in November - once they sat within the drop zone.More news: Ocala teacher accused of drowning raccoons during class
Sullivan has reportedly already held talks with Shakhtar Donetsk manager Paulo Fonseca, but reports from Fonseca's homeland Portugal claim he has instead opted to sign a new contract with the Ukrainian side.
"Throughout his time here, David has carried himself with dignity and honesty and we have all found him to be a pleasure to work with".
"They deserve great respect for the job they have done and they leave the Club with our best wishes".
On Tuesday fellow co-chairman David Gold had claimed he wanted Moyes to stay but in a statement on Wednesday Sullivan said the club had chose to "move in a different direction". Fans have been increasingly disenchanted with the club's owners and protested at some home games.
Moyes said: "I could have joined a Premier League club during the season here, when I was West Ham manager".
Former Manchester City boss Manuel Pellegrini, now with Hebei China Fortune, outgoing Paris St Germain coach Unai Emery, and Newcastle's Rafael Benitez, have also been linked with the role.