Officials said the attack happened at the Europa shopping centre in the capital Minsk.
Eyewitnesses spoke of a masked man chasing terrified shoppers on Saturday evening.
Police later arrested the 18-year-old suspect. The motive for the attack was not immediately clear.
"When he took out a chainsaw, shoppers first thought it was some kind of game," police spokesman Alexander Lastovsky told Reuters.
"Then he cut one woman badly with the chainsaw and hit her with a hammer. She died. He also attacked others."
The injured woman was taken to a local hospital.
The attacker had brought the chainsaw and the axe to the shopping centre in a guitar case.

Japanese-owned chemical firm Sekisui Alveo said production at its Merthyr Tydfil Industrial Park plant will end by March 2018.
The plant will be decommissioned in four stages over the next three years, the company said.
It is part of a move to relocate production to Roermond in the Netherlands, it added.
Alan Gunter, manufacturing director at the Merthyr Tydfil site, said the decision to decommission the plant had "not been taken lightly".
"I am immensely proud of our highly professional workforce and I would like to thank them all for their commitment and dedication," he said.
"We are aware that this is a difficult time for employees at the Merthyr Tydfil production plant and we will do everything to support them and their families during and after the plant's staged decommissioning.
"We will also continue to collaborate with the local authorities in the coming months."

John Ridsdel, 68, was taken from a tourist resort along with three others by the Abu Sayyaf group in September last year.
The group had threatened to kill a male hostage on April 25 if a ransom was not paid.
Confirming the death, Canadian Prime Minister Justin Trudeau called it "an act of cold-blooded murder".
Mr Ridsdel was kidnapped from a marina near the city of Davao along with another Canadian, Robert Hall, Norwegian Kjartan Sekkingstad and a Philippine woman, Mr Hall's girlfriend Marites Flor.
They were taken 500km (300 miles) to the island of Jolo. Abu Sayyaf released a video of the group in November, in which they demanded $80m (Â£55m) for their release.
Mr Ridsdel later warned that he was due to be killed if no ransom was paid.
Several hours after the deadline, a severed head was found in a street on Jolo. The Philippine authorities said it belonged to a foreign man but it has not yet been formally identified.
"It's hard," a friend of Mr Ridsdel, Bob Rae, told CBC News. "It's just very hard. I've been involved behind the scenes for the last six months trying to find a solution and it's been very painful."
A former mining executive, Mr Ridsdel is described by Canadian media as semi-retired.
He also worked as a journalist.
Offering his condolences, Mr Trudeau gave few details, saying he would not compromise the safety of the other captives.
Abu Sayyaf was set up in the 1990s and received funding from al-Qaeda.  It is fighting for an independent Islamic province in the Philippines.
Several of its factions have declared their allegiance to the militant group Islamic State. Abu Sayyaf is also holding several other foreigners.
Eighteen Philippine soldiers were killed in clashes with the militants on Basilan island near Jolo island earlier this month.

Gordon Smith, 44, was found dead on Lake Road about 400m (0.2 miles) from his home on Church Street on 1 July.
Mark Russell, 27, previously of Holly Terrace, pleaded not guilty to manslaughter at Carlisle Crown Court.
He was was given conditional bail ahead of trial on 14 March. It is expected to last up to five days.
Judge Paul Batty QC said one of the conditions is that he does not enter Windermere or Bowness-on-Windermere.
A 31-year-old man from Millom, a woman aged 48 and a 15-year-old girl, both from Kent, previously arrested on suspicion of murder, have been released with no further action, Cumbria Police said.
A young man has attacked people with a chainsaw and an axe at a shopping centre in Belarus, killing one woman and injuring another.
[NEXT_CONCEPT]
A foam factory in Merthyr Tydfil has announced it will close with the loss of 80 jobs.
[NEXT_CONCEPT]
A Canadian man held captive by Islamist militants for months in the Philippines has been killed.
[NEXT_CONCEPT]
A man has denied the manslaughter of another man who was found lying in a street in Windermere close to where he lived.