People in the constituency will now vote on May 28 and counting will be held on May 31.
The election was deferred due to various irregularities, the state electoral authorities said.
The decision comes days after nearly 10,000 voter identity cards were recovered from a flat in the constituency, leading to a slugfest between the Bharatiya Janata Party and the Congress.
The order, signed by the Chief Election Commissioner and two fellow commissioners, said 'the Commission is satisfied that the current electoral process (in the constituency) has been vitiated on account of unlawful activities of candidates and political parties'.
It said the parties and candidates tried to woo the electorate to vote for them and also tried to keep them away from casting their ballot.
The poll watchdog also directed the election authorities in the state to provide fresh voter identity cards (Electoral Photo Identity Card or EPIC) to all those whose cards were recovered from the flat seven days prior to the fresh election date.
The EC order said the cards belonged to 'voters in the vulnerable community'.
It said random checks found that the cards were genuine and further investigation was required to ascertain who broke the door of the flat before three police constable reached the scene after receiving a phone call.
The order also quoted a ground report by deputy EC which said that there is a 'design to induce voters' and that the planning was on even before the elections were announced which was a 'serious cause of concern'.
Using its powers under Article 324 of the Constitution to postpone the polls in R R Nagar, the EC said there is a 'definite inference that all the above seized gift items and collection of EPIC were likely for large-scale inducement of the voters'.
It pointed out that bribery and inducement of voters is an offence under the Indian Penal Code.
Besides this, the commission took a serious view of the discovery of a huge cache of water cans, tee-shirts, half-pants and pressure cookers meant to bribe the voters, he said.
Chief Electoral Officer Sanjeev Kumar said discovery of these materials was a serious offence under various sections of laws. It comes within inducement under various sections of the Indian Penal Code, as well as Representation of the People Act, he added.
Conviction of such persons, specifically if under Section 121 of the RPA, can result in the election of the returned candidate being declared void, he said.
Besides, the candidate found guilty of such a corrupt practice can also be disqualified by the President on the recommendation of the commission, said Kumar.
He said the investigation led to the recovery of sarees and water cans from houses. The cans were similar to what the commission found in Jalahalli apartment, where the voter ID cards were found.
"It appears that attempts were made to systematically influence voters," Kumar said.
He said besides water cans, EPIC, pressure cooker, a truckload of 5,018 tee shirts with photos of the candidates and 23,393 half-pants were found.
A truck was dispatched from Tirupur to Tumakuru, but was unloaded at Bengaluru.
While Reliance Retails came forward to claim the half-pants, nobody came to claim the tee-shirts, Kumar said.
The tee-shirts carried the photo of the same candidate who did the survey and other things were quite visible, he said.
The BJP and the Congress, the two main contenders for power, went after each other claiming that the flat -- from where the cards were recovered -- belonged to a woman associated with the rival party.
The BJP said the EC order deferring polls and registration of a case against the Congress candidate was a 'clear indictment' of the ruling party in the state and proved that it was 'using money power to influence the elections'.
On the eve of the polling, Union minister Piyush Goyal, in a scathing attack on the Congress, alleged that the ruling party had resorted to corrupt practices to rig the elections.
He also said the BJP had asked the EC to seek one more proof of address from first-time voters, besides the voter ID card, to ensure that they are genuine voters.
"A desperate Congress has given up all hope of winning the elections and is resorting to most corrupt electoral practices to rig the state polls," Goyal told reporters in New Delhi.
The minister termed the EC order postponing the polls to the R R Nagar constituency an 'extraordinary' step.
"The EC and its special observers had detected specific instances of corrupt electoral practices and attempts at bribery. The poll panel has taken an extraordinary step to postpone the election in the Rajarajeshwari Nagar seat," Goyal said.
"We have also asked the EC to put out an advertisement asking voters not to get carried away by any inducement or coercion, and exercise their right in a free and fair manner," he said.
While the seizure was made in one seat, similar methods may have been adopted in other seats. The EC should conduct a probe across the state, he said.
"The seizure of 9,564 EPIC cards, photo voter slips, laptops, and other materials indicates there was a clear attempt to induce voters in a systematic and planned manner... it vitiates the election process," Goyal said citing the EC order.
He said the BJP had also asked the EC to take note of a similar malpractice in a neighbouring constituency, and alleged that the Congress was distributing money in Bedami, one of the seats from where Chief Minister Siddaramaiah is contesting.
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