The comments by Modi were his first on the subject since the leaking of question papers for a national entrance test to medical school in May, the anger over which has become the biggest challenge for the prime minister in his third term.
Thousands of youngsters marched on parliament on Monday demanding the resignation of Education Minister Dharmendra Pradhan over the leaks that affected more than two million students and led to some dying by suicide.
Delhi Police said nearly 180 people, including 118 security and police personnel and 60 protesters, were injured on Monday when they clashed in central Delhi.
The founder of the self-named Cockroach Janta Party (CJP) said more than 150 protesters were undergoing treatment in hospital.
Modi told MPs of his ruling alliance that the government took immediate action on receiving reports of exam irregularities and 13 individuals were arrested and jailed, Parliamentary Affairs Minister Kiren Rijiju said.
Modi also said a re-examination was successfully conducted and there was no delay in declaring their results, Rijiju said.
"The prime minister said ... that to ensure the future of youth is not jeopardised ... everyone should unite to ensure strict steps are taken, those guilty punished, and a foolproof system is created," Rijiju said.
Modi's comments came as the CJP movement said it would continue its anti-government protest but would not march on parliament because it feared its supporters would be beaten by police again.
Thousands of protesters from Delhi and nearby areas had gathered to march on parliament on Monday, joining a movement that began as an online satire.
The months-old movement by the CJP has drawn the support of millions of young, gen Z Indians who are demanding the resignation of Pradhan.
CJP says the leaking of the papers in May was a sign of deep-seated corruption in education.
Nearly 500 protesters gathered at the Jantar Mantar protest site in central Delhi on Tuesday amid tight security.
CJP founder Abhijeet Dipke said the protesters would not resume the march to parliament as the group did not want more youth to be hurt.
Delhi Police said protesters had "displayed unruly, aggressive and violent behaviour" and "deliberately violated the prohibitory orders in force", despite warnings on Monday.
The movement galvanised after the support of activist Sonam Wangchuk, who began a hunger strike on June 28, but was forcibly moved to a hospital by police on Saturday.
On Tuesday, the New Delhi High Court allowed the Magsaysay award winner to be shifted to a private hospital from the state-run hospital where was taken following a petition from his wife that alleged he had been illegally detained.
