Emergency services were called to the scene in the east of the city just after 2.20pm on Tuesday afternoon.
Police told the TT news agency shortly afterwards that it was not clear how seriously the woman had been injured, but that she remained in hospital.
But a witness told Sweden's Aftonbladet newspaper that they had seen "blood flowing and people around [the area] were in a panic". The bystander added that they had seen a man crying next to the victim.
The shooting took place just outside two adult education centres and the witness – a male student – added that he and his classmates had rushed outside to see what was going on.
"She just lay there on the ground. Opposite our school is a medical training centre, so a few nurses and one doctor came from school with towels and tried to stop the bleeding," he said.
The witness was understood to be a pupil at Lernia Vuxensgymnasium, an adult education centre which teaches Swedish as a foreign language, as well as vocational courses.
The area around the street where the shooting took place (Anders Personsgatan) was cordoned off and police were carrying out a forensic examination of the site on Tuesday evenning.
The shooting took place a week after a deadly attack in Biskopsgården, a different suburb in the west of the city and police were quick to suggest that they did not think the events were linked.
"There is currently no evidence to suggest that this would have something with gang conflicts", Jenny Widen, a spokesperson for Gothenburg police told TT.
A suspect has been arrested.