Moyes was mysteriously absent from the touchline during the start of the tie in east London, while there were two late changes to the squad with both Diop and Cullen dropping out.
The club have confirmed that they have tested positive for Covid-19 and were sent straight home from the stadium.
Despite the positive tests, the game has still kicked off with assistant manager Alan Irvine taking charge of West Ham.
‘West Ham United can confirm that David Moyes, Issa Diop and Josh Cullen have returned positive tests for COVID-19,’ a statement read.
‘The club’s medical team were informed of the test results as the team were at London Stadium preparing for this evening’s Carabao Cup third round tie against Hull City.
‘The manager and both players immediately left the stadium and have returned home.
‘All three, who are asymptomatic, will now follow Public Health England and Premier League guidelines and protocols.
‘Everyone at West Ham United send their best wishes to David, Issa and Josh.
‘The Club’s measures and protocols around COVID-19 remain stringent. This included offering to test the Hull City team ahead of tonight’s fixture – an offer which they opted not to accept.’