Comedy Central cancelled Wednesday's live show and aired a repeat, but said "the procedure went well" and Noah was "currently recuperating from surgery".
"We expect he and the show will be back with an original episode tomorrow," said a statement on Twitter.
The South African replaced Jon Stewart, who stepped down after 16 years.
Actress Regina King, star of US shows American Crime and The Leftovers, is lined up to be Noah's guest   on Thursday, said Comedy Central.
Fellow late night TV host Jimmy Fallon, of The Tonight Show, was one of the people who wished Noah well on Twitter.
He tweeted: "Hey @Trevornoah, tell everyone at the hospital I said hey! Get better soon buddy!"
Before taking over as host on The Daily Show, Noah, 31, made his debut as an international correspondent in December.
Noah appeared on three different occasions on The Daily Show before being offered the top slot. His first show on 28 September received mixed reviews from critics.
The Hollywood Reporter wrote:  "No memories of Stewart and his storied reign have been usurped or replaced. Nor, however, have they been sullied."
Deadline called the evening "mostly inoffensive but also mostly flat".
But during the first four weeks as host, audience figures rose 20% among young adults aged 18-24.
Noah is the third host of the nightly show. Prior to Jon Stewart,  Craig Kilborn was the first face of the series from July 1996 to December 1998.
Noah, the son of a black South African woman and a white Swiss man, comes from South Africa's biggest township of Soweto.
He is a popular comedy star in his home country largely due to his humour cutting across the racial divide.
Trevor Noah, who took over as host of The Daily Show in September, has undergone emergency surgery to removed his appendix.