Khan, 65 – likely to fall short of a majority and expected to rule in a coalition – said: “We were successful and we were given a mandate.”
Campaigning was marred by violence – with a bomb killing 31 people at a polling station.
Oxford University graduate Khan, who captained Pakistan to World Cup victory in 1992, had come under fire over claims his bid was supported by military interference in the nuclear-armed republic.
But he hit back at claims of corruption by former prime minister Nawaz Sharif, saying: “I think this has been the clearest, fairest election Pakistan has ever had.”
Khan was a key part of London’s high society in his younger days, at one time married to Jemima Goldsmith.
The journalist, whose brother is Tory MP Zac Goldsmith and who has two sons with Khan, was full of praise for the ex-cricketer’s “tenacity, belief and refusal to accept defeat”.
Congratulating him, she tweeted: “After humiliations, hurdles and sacrifices, my sons’ father is Pakistan’s next PM.”