News agency TT reports that the man had been armed with a knife.
Just before 7pm, the police say a man attacked a woman with a knife, and that after trying to stop him by using pepper spray and a baton they fired at the man.
"When we get to the scene, the patrol sees through the window of a home that the man is attacking the woman with a knife," Joakim Hansson, the deputy inspector on duty with the Västra Götaland police, told TT.
The police tried to ward off the attack with a baton and other means, but when that did not work, they fired against the man, and the shot went into his abdomen.
"This was their last resort, to save her life," says Lena Matthijs, district police commissioner and area chief in Älvsborg, said.
The man is reported to have cut the woman several times, including her head and back, before he was shot. Both of them were taken to the hospital, and the man died at around 11pm.
The woman has been operated on and is reported to be in serious, but not life-threatening, condition.
This makes the eighth deadly shooting by the police in two years, reports TT.
The Swedish Police Union is now reiterating its demand that Swedish police officers are allowed to be equipped with electroshock weapons. Chairperson Lena Nitz tells Swedish Television News that they believe some of these deaths could have been avoided, had the police been allowed to carry electroshock weapons.