A spokeswoman for the school has told the New Zealand Herald tonight's show as well as two other performances originally planned to run over the weekend would not be going ahead.
Tickets to tonight's show were still available this morning for $36 with a $6 booking fee.
One ticket holder said they had not yet received confirmation the show had been cancelled.
Two boys were taken to hospital on Wednesday, the performance's opening night, after their necks were cut with a real razor wrapped in duct tape which was used as a prop.
Both boys were discharged from hospital by yesterday afternoon, and school headmaster Steve Cole said he understood they were doing well.
However, he admitted the school was still in the dark as to how the incident could have occurred, saying the antique blade had been made safe.
The prop was a razor that had been blunted and wrapped in several protective layers, Mr Cole said.
"It's normal for Sweeney Todd to have such an instrument [and it] clearly had been checked many, many times," he said.
The unfortunate incident happened halfway through the second act, Mr Cole said.
The performance continued and audience members were unaware of the incident.
Yesterday Mr Cole said he hoped the rest of the shows in the run would be able to go ahead.
The school made the decision to cancel the shows late this afternoon.
The prop supplier reportedly warned St Kentigern College not to use a real cut-throat razor.
As authorities at the private Auckland school try to figure out how the incident happened, the prop supplier told Fairfax that members of the production had come looking for real razor blades.
"They were looking at the real razor blades ... we've got some rusty ones here ... we don't let school productions have those."
The scene had been performed "many times" since January, including eight dress rehearsals where the prop was used.
When asked how two boys ended up with the sides of their necks cut Mr Cole last night said he didn't know.
"I've got no idea. I've got no idea."
Senior staff at the school spent much of yesterday in crisis meetings, and a police investigation is under way.
"It's not a criminal inquiry. In this instance the play is not the reality," said a police spokesman.
WorkSafe New Zealand is also looking into the incident, but Mr Cole said he was not concerned about possible consequences for the school under new health and safety laws which came into force on Monday.
"We're very confident in our health and safety procedures.
"My concern is for the two boys and their families."
The owner of an antique shop where the razors were bought said the person who purchased them had talked to one of his staff members about the need to cover up the sharp edge.
The man, who asked the Herald not to name him, could not immediately recall whether the man had mentioned it would be a high school production.
"I know they discussed the safety element and maybe covering up the blade."
Sweeney Todd is about an English barber and serial killer who kills his customers with a razor and, with the help of struggling pie shop owner Mrs Lovett, turns their corpses into meat pies.
In 2007 the musical was made into a film starring Johnny Depp and Helena Bonham Carter.
