The local governor's office said the attack, which took place in the morning in Izmir's Konak district, was carried out by a person with mental health issues.
The police are investigating the incident, it said. The governor's office did not mention the consulate.
But broadcaster NTV said the attack took place outside Sweden's honorary consulate. The wounded woman, who was working as a secretary at the mission, was in critical condition, it said.
Turkish authorities detained the assailant and have launched an investigation into the incident, the governor's office said.
Honorary consulates represent nations' interests abroad but are not run by professional diplomats.
The Swedish consul general will travel to Izmir on Wednesday to get more information about the incident, the Swedish Foreign Ministry said.
It said it was in close dialogue with the consulate general in Istanbul and its staff in Turkey.
The ministry said it would not comment further "on the threat to the diplomatic service or what security measures are being taken, as this risks counteracting the purpose of the measures".
Turkish Justice Minister Yilmaz Tunc condemned the attack in a post on messaging platform X, formerly known as Twitter.
Mr Tunc said a criminal investigation had been started into the incident.