The worst political crisis of Modi's third term in office had peaked this week when thousands of protesters marched on parliament, with police using tear gas to control them.
Young voters form a critical base of his Bharatiya Janata Party.
But youths calling themselves "cockroaches" had taken to the streets since late June after exam papers were leaked forcing a re-sit which affected two million students and galvanised anger about the lack of prospects for young Indians.
The resignation of Education Minister Dharmendra Pradhan, the students' primary demand, came as a surprise because Modi has a reputation for refusing to act under pressure.
Modi's party faces key state elections next year.
Similar youth-led protest movements brought down governments in neighbouring Sri Lanka, Nepal and Bangladesh in recent years.
Celebrations erupted at the Jantar Mantar protest site in the capital Delhi, with thousands of youngsters chanting victory slogans and dancing.
The youth-led Cockroach Janta Party (CJP) movement, which spearheaded the demonstrations, said it would end its protests.
"We have done it," CJP founder Abhijeet Dipke said to loud cheers at the site.
Youngsters shouted "Jai Hind," a patriotic slogan meaning "Victory to India," as the national anthem played on loudspeakers and sweets were distributed.
The CJP was founded in May after a chief justice referred to unemployed young people as cockroaches.
It describes itself as a "satirical political movement" and organises its activities primarily online.
Roads around the central business district of Connaught Place near the protest site were shut on Saturday amid celebrations and a swelling crowd of thousands of protesters.
Health Minister JP Nadda and Jitendra Singh, a junior minister in Modi's office, held a joint press conference with two CJP spokesmen after another round of talks following Pradhan's resignation.
Nadda said the government had accepted all demands of the protesters, which included reforms in the exam system, dropping police cases filed against protesters and compensating families of students who died by suicide after the exam paper leaks.
"The Cockroach Janta Party declares that we withdraw the agitation in good faith," CJP spokesman Saurav Das said.
"All the demands raised by our team have been accepted, therefore we request protesters to withdraw and peacefully go home," spokesman Ashutosh Ranka added.
"The bottom line is we have won, your struggle, hard work, and patience have paid off," Ranka told cheering crowds at Jantar Mantar on returning from the talks, and urged them to withdraw peacefully.
Opposition parties had echoed the demands of the youth and disrupted the monsoon session of parliament for the entire week, threatening key legislation planned by the government.
Scenes of police using tear gas to control protesters had also fuelled public anger, although authorities defended their actions saying they were only enforcing the law.
Pradhan announced his resignation in a post on X.
"Considering the situation that has arisen at Jantar Mantar and across the country, so that anti-national forces do not take advantage of this situation ... I have sent my resignation letter to the prime minister," Pradhan wrote.
"I deeply respect the aspirations, feelings and legitimate expectations of the country's youth."
The protesters had demanded his resignation over leaks of the medical school entrance test paper in May, which led to the cancellation of results and a retest which affected two million students.
The protests also reflected growing anger in India over job scarcity and government accountability.
Pradhan's resignation is "a victory of peace, patience & perseverance," activist Sonam Wangchuk, who was on a hunger strike for 26 days to support the youth movement, wrote on X.
"Young generation is here. This is a win for the constitution," protester Tluanga Ralte said, walking with a copy of the Indian constitution at the protest site after Pradhan's resignation.
"We are not a lazy generation."
with DPA
