His son Elie wrote: "Despite the NHS doing an amazing job trying to save him, he unfortunately didn't make it.
"I don't really know what to say other than my family and I are completely devastated.
"Not only did he mean a lot to us. He also meant a lot to wider Jewish community."
Rabbi Kraft's son asked that the community now allow the family "some space to grieve."
The JC understands that the rabbi had an underlying health issue before contracting coronavirus.
The rabbi had previously led the Woodford and District Liberal Synagogue community and South London Liberal shul.
Born and raised in Boston, Massachusetts, he had two sons with wife Susannah.
In a statement the Board of Deputies said they "share the deep sorrow of many in the community".
The Board added: "We wish every comfort to his wife Susannah and his two sons. May his memory be for a blessing."
