Thirty-five-year-old Shah and his Rastriya Swatantra Party were on course to win a majority in the 275-member lower house of parliament, leading in 100 of the 137 seats, the election commission said.
Vote counting was under way on Friday for 165 seats filled by direct election in the 275-member lower house while the rest will be decided through proportional representation.
Shah, the former mayor of the capital Kathmandu, has dominated the race to become prime minister, gaining near-rockstar-like fame on social media across the Himalayan country after a youth-led uprising.
"At least one member from each household seems to have voted for it (RSP). Otherwise this type of tsunami would not have been possible," analyst Puranjan Acharya told Reuters.
Political instability has plagued the country of 30 million for decades, crippling a largely agrarian economy grappling with unemployment and corruption.
That unrest erupted into street protests last September after a social media ban brought thousands onto the streets, triggering clashes and deaths that led to former prime minister KP Sharma Oli's resignation.
If Shah takes power, it would cap a dramatic rise for a man who first gained prominence with rap music critical of the establishment and is now seeking high political office.
His songs, often taking on Nepal's ruling class, struck a chord with many in a country where about 20 per cent of the 30 million population live in grinding poverty.
Released in 2019, one of Shah's best-known songs, Balidan - or sacrifice in the Nepali language - has more than 12 million views on YouTube.
Its lyrics read:
"Let me speak, sir, it is not a crime,
Let me open the mind, I am not a curse to the palace,
My mind is not bad, it is not afraid to speak the truth."
Trends showed Oli's Communist Party of Nepal was leading in 10 seats and he was losing to Shah in his home constituency.
The Nepali Congress, the country's oldest party, was leading in 10 seats and had won just one.
Vice President Bishwa Prakash Sharma said in a post on X that the party had accepted the voters' verdict.
"I voted for Rastriya Swatantra Party because Balen is there," said Deepak Adhikari, 33, after voting in Kathmandu on Thursday, referring to the youthful leader by his popular name.
"I believe he will ... make the country prosperous."
Final results were likely to be announced on Friday evening or by Saturday morning, election officials said.
