Hammers manager Moyes had named Diop and Cullen in his line-up for the third-round match on Wednesday (AEST) but the trio returned home upon learning of the results to being self-isolating.
The match started as planned, with Harrison Ashby and Jack Wilshere promoted to West Ham's starting XI and assistant manager Alan Irvine overseeing the team.
A statement read: "West Ham United can confirm that David Moyes, Issa Diop and Josh Cullen have returned positive tests for COVID-19.
"The club's medical team were informed of the test results as the team were at London Stadium preparing for this evening's EFL 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."
Tottenham had its EFL Cup fixture called off on Wednesday after Leyton Orient revealed "a number" of its players had tested positive for coronavirus.