Since the Bharatiya Janata Party did not put up any candidate due to lack of numbers in the assembly, Singh was declared elected after the scheduled time to withdraw nominations ended on Monday.
The seat had fallen vacant due to the death of sitting member, Madan Lal Saini of the BJP.
Singh was set to be declared elected unopposed on the last day of withdrawal of nominations on August 19 as the opposition BJP had decided not to nominate a candidate for the bypoll.
Singh had filed his nomination papers on August 13, accompanied by Chief Minister Ashok Gehlot, Deputy Chief Minister Sachin Pilot, AICC general secretary in charge for Rajasthan Avinash Pande, Parliamentary Affairs minister Shanti Dhariwal, chief whip Mahesh Joshi, Health minister Raghu Sharma and others in the chamber of the returning officer and assembly secretary Pramil Kumar Mathur.
Later, he told reporters that he will do whatever is possible to promote the cause of the people of the state.
He also condoled BJP leader Madan Lal Saini's death in June, which led to the vacancy in the upper house.
"I am very grateful to the Congress party and Rajasthan Chief Minister Ashok Gehlot, Deputy Chief Minister Sachin Pilot and all members of the Congress party for nominating me to this vacant seat in the council of states," he told reporters after filing the nomination.
Saying he recognised there were circumstances in which the vacancy was created, he conveyed his condolence to Saini's family.
"I express gratefulness to the people of Rajasthan and the Congress party for giving me this privilege for serving the people of Rajasthan in the council of states. I will do my very best to do whatever is possible to promote the cause of the people of Rajasthan," he added.
Pilot said the Rajasthan Congress will benefit from Singh's experience.
The Congress has a majority in the state assembly and Singh, thus, is likely to win the bypoll.
The Rajasthan Assembly has 200 seats, but two seats are vacant.
Singh, the veteran Congress leader, was a member of the upper house from Assam for almost three decades.
He was a Rajya Sabha member for five consecutive terms from 1991 to 2019 and the country's prime minister for two consecutive terms between 2004 and 2014.
His tenure in Rajya Sabha ended on June 14 this year and he could not be renominated as the Congress party did not have the strength to send him to the upper house from Assam again and there was no other vacancy left.
Rajasthan has 10 Rajya Sabha seats and one got vacant after the demise of Madan Lal Saini, who was also the state president of the BJP.
The remaining nine seats of Rajya Sabha from the state are with BJP.