Six of them are in critical condition, while others suffered from burns and smoke inhalation - including children.
Police said the suspect walked into a Commonwealth Bank branch in the suburb of Springvale and used an inflammable liquid to start the fire.
The man was taken to hospital for treatment while under police guard.
Acting inspector Jackie Poida from Victoria Police told reporters the 21-year-old suspect "had with him some sort of accelerant, he walked into the bank and he lit that accelerant which caused some sort of fire... He set himself alight with that fire."
She said it was "too early to tell" what the man's motives were. The suspect is among those critically injured.
The incident took place at about 1130 local time (0130 GMT).
Video taken by eyewitnesses of the immediate aftermath showed flames inside the bank branch and thick black smoke, according to the Australian Broadcasting Corporation.
Another eyewitness said he heard a "huge blast" and saw people running out with severe injuries.
Ashley Atkin-Fone said he took a fire extinguisher from a nearby shop and used it to put the fire out.
"I took four steps inside, I was shouting and shouting, I couldn't hear anyone," he told reporters.
Authorities said that those being treated for their injuries include a toddler and elderly residents.
An Australian man has started a fire at a bank in Melbourne, setting himself alight and injuring at least 26 people, police said.