Ms Swaraj's assurance to the family from Lahore came after the mother of the child, Rohaan, sought the minister's intervention.
"We will give medical visa for his treatment in India," Ms Swaraj tweeted.
A number of Pakistani citizens had tweeted requesting India to issue medical visa for the child's treatment.
"When I hug my child, he smiles. He knows he is safe with me. I want to save him, I really do. Help in visa #HelpBabyRohaan @SushmaSwaraj," Rohaan's mother Mehwish Mukhtar had tweeted.
A Pakistani national Fawad said Rohaan is a newborn and has a heart problem.
In recent months, Sushma Swaraj has heeded to requests by a number of Pakistani citizens in granting medical visas despite strain in ties between the two countries due to a host of issues.
On August 15, Ms Swaraj had said India would grant medical visa to Pakistani nationals in genuine pending cases. "On the auspicious occasion of India's Independence day, we will grant medical visa in all bonafide cases pending with us. @IndiainPakistan," she had tweeted.
In May, the external affairs minister had said that India would grant medical visa to those nationals from the neighbouring country whose applications were recommended by Pakistan Prime Minister's Foreign Office Advisor Sartaj Aziz.
Mr Aziz, however, is no longer the Pakistan PM's foreign office advisor as he has been appointed as the deputy chairman of the planning commission.
Shahid Khaqan Abbasi Abbasi, 58, was sworn-in as Pakistan's 18th prime minister on August 1 after the Supreme Court disqualified the incumbent, Nawaz Sharif, for dishonesty following the Panama papers scandal.