The attack was the latest in a string of violent, often deadly assaults in Europe in recent months, with many claimed by the jihadist Islamic State group.
The motive for today's attack was not immediately clear, police told AFP.
"According to the information we have for the time being, the 27-year-old Swiss man poured out a flammable liquid ... (which) caught fire," police said, adding that the man "was also armed with at least one knife".
Police said the attack had taken place at around 2:20 pm (local time) near the Salez station on a train running between Buchs and Sennwald and that some of the injuries were serious.
Seven people including the suspected attacker were admitted to various hospitals with burn and stab wounds, the statement said.
The victims included two men aged 17 and 50, three women aged 17, 34 and 43 and the six-year-old child. Dozens of people were on the train at the time of the attack, the police statement said.
The Salez station remains closed, and police said replacement buses had been set up.
Saint Gallen prosecutors have opened a criminal investigation into the attack.
The attack is the latest in a string of brutal assaults in Europe in recent months that have left the continent on edge.
In January 2015, a jihadist assault on the Charlie Hebdo satirical newspaper in Paris left 12 dead while another four died in an attack on a Jewish supermarket in the city.
In November 2015, a coordinated jihadist attack claimed by IS in Paris left 130 people dead.
On July 14, a Tunisian man rammed a truck into crowds of revellers in the southern French city of Nice, killing 85 people.
Swiss neighbour Germany has also seen a string of attacks in recent weeks.
In July there were two attacks by migrants in the southern state of Bavaria -- an axe rampage on a train and a suicide bombing.
In Munich, an 18-year-old German-Iranian also went on a gun rampage in a shopping mall, leaving nine people dead.
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