The 31-year-old Austrian who won his country’s parliamentary elections this week by poaching votes from far-right nationalists now faces the test of figuring out how to govern with them.

Austria’s head of state on Friday formally invited Sebastian Kurz, who led his Austrian conservatives to first place in parliamentary elections on Sunday after copying the anti-immigration stance of the country’s far-right Freedom Party, to form a new government. Mr. Kurz is widely expected to do so by forging a coalition with the far-right...