Vučić had spent two terms as president and could not seek a third term, so was leading his Serbian Progressive Party (SNS) with the aim of returning to the post of prime minister, which he left in 2017.
The Student List's lead candidate Ilija Srdanović told BBC Serbian: "This will be a free state."
He said the anti-corruption movement, which had attracted anti-government voters across Serbia, hoped to "free the country from tyranny".
There were celebrations on the streets of the city, with people dancing and chanting "victory", cars honking, and fireworks set off. "Serbia is free," one student told the BBC. "It's a miracle."
During a celebration speech, Student List candidate Radomir Saicic said Serbians had shown "no obstacle was greater than their will", Reuters reported. "No force can forever overpower a people who have decided to take their destiny into their own hands," he added.
Amid scenes of people dancing in the streets Belgrade, Selena Lebovic said the result was "really fabulous".
"We are all crying, and what to say. We couldn't believe that it would be so easy," she told Reuters.
The anti-corruption movement campaigned on a manifesto that included purging officials who broke rules cracking down on protesters, stopping party loyalists from leading public enterprises, and freeing the public broadcaster of political control.
Vučić and the SNS party have repeatedly denied corruption.
Vučić, who backs Serbia's candidacy for the European Union while refusing to impose sanctions on Russia over its full-scale invasion of Ukraine, had pledged to increase pensions and public sector salaries, and investment in state healthcare.
Preliminary figures provided by Serbia's election commission suggested a record high turnout, with 64.9% having cast their vote by 18:00 local time (17:00 GMT), two hours before polls were set to close.
Earlier in the day, long queues were seen outside polling stations in cities and towns across Serbia.
This was also the case outside booths in a number of countries with sizeable Serbian communities - including Slovenia's capital Ljubljana, Germany's Munich, and Los Angeles and Chicago in the US.
In France, Serbian voters waited in line in heavy rain in Paris.
The election was held amid high tensions across Serbia, with police sealing off streets around the Republic Electoral Commission (RIK) in Belgrade.