An official statement read out on national radio said: "The security conditions are not sufficient in Anjouan to allow the elections to take place this Sunday."
It also said all three islands will hold the second round of their ballots on June 24.
Peaceful and transparent
An Anjouan government spokesman rejected the decision and said Sunday's poll would go ahead as planned.
But Ahmed Ben Said Jaffer, the federal foreign minister, said no election materials had been sent to the island yet.
"I don't see how the island's government can organise these elections," he said, adding that four of Anjouan's six presidential candidates had also asked for the one week delay.
In a statement on Friday, the European Union called on all sides to ensure the elections were peaceful and transparent.
Three people were wounded on Tuesday when Anjouan police opened fire on a crowd waiting to meet a flight carrying the Comoros' federal president, Ahmed Abdullah Mohamed Sambi.
The runway was blocked and Sambi forced to turn back.
Tensions rose sharply on the island in May when forces loyal to its then-leader Mohamed Bacar killed two federal soldiers.
An envoy from the African Union, which brokered an end to that fighting, condemned Anjouan's leadership on Wednesday, saying the blocking of Sambi's flight had been "intolerable".
The three islands, whose total population is about 670,000, retain autonomy via local presidencies under the terms of a 2001 peace deal and share a rotating national president.
Sambi's election last year as federal president was the islands' first peaceful transition of power since independence.
First settled by Arab seafarers 1,000 years ago and later a haven for pirates pillaging ships across the Indian Ocean, the rocky islands were annexed by France in 1904.