Imran Khan is a newcomer to power in Pakistan, but his first challenge is a familiar one for the country: a crushing financial crunch that will likely require an international bailout and spending cutbacks.

After decades on Pakistan’s political fringe, nearly complete results show that Mr. Khan’s party took more than twice as many seats in parliament as its main competitor, whose highest profile leader is the now-jailed former Prime Minister Nawaz Sharif.

Mr....