The passengers, all of whom were in their early 20s, were on a helicopter ride organised by the Høstsprell music festival.
"What has happened is deeply tragic and we are all in shock," the festival wrote in a post announcing that concerts would be cancelled on Saturday evening. "The festival management's thoughts go out to all the relatives."
Mari Strifeldt Arntzen. a local, told Norway's VG newspaper that she had been one of the first to report the accident.
"We first saw a lot of black smoke and flames, so we first thought it was a fire. But we got a jolt when we realized what it was and that there were people involved. It's very sad," she said.
Witnesses told the local Altaposten newspaper that they had heard five or six small explosions.
Only one of the passengers had been alive at the time rescue services arrived, although they too had died shortly after arriving at hospital in Tromsø, Norwegian police said in a press release on Sunday morning.
Another was initially thought to be missing, but their body was found in the wreckage at 8.40pm.
"The crashed helicopter burned intensely after the accident, and it should be expected that it will take some time to recover and identify the dead," Gunnar Øvergaard, Operations manager for the local Finnmark police, said in the statement.
"The police have requested assistance from the identification group within Kripos [Norway's crime investigation body] in this work. They will begin work on Sunday."
Jonas Beltrame-Linné, a press secretary with the Swedish police, told the Expressen newspaper that the pilot's relatives had been informed of his death.
The crash was first reported at around 5pm on Saturday and police found the sixth body at 8.40pm.
Many other festival goers had used the helicopter, which was operated by the company Helitrans, to get to the festival.
"The journey took seven minutes and we felt safe the whole time and felt that the pilot was in control," Jens Karlsson told the Dagbladet newspaper.
"The helicopter took a tour around central Alta and over a chain of mountains before it landed again."
"Then we got out and let the next lot of passengers take our places. It was them that crashed. It's pretty crazy to think about it."