The first attack near the Nestroyplatz metro station resulted in a couple and their 17-year-old daughter receiving severe injuries.
The suspect, an Afghan national, blamed his "aggressive feeling and rage about his life situation" for the stabbings, according to police spokesman Patrick Maierhofer.
The 67-year-old father is still in a critical condition, while the injuries of his two relatives are not thought to be life-threatening, police said.
Around half an hour later the man attacked another Afghan man in the Praterstern area.
Maierhofer said the suspect knew the man and blamed him for his drug addiction problems.
Police said the victim is not thought to be in a life-threatening condition.