The island, however, rejected the order despite the African Union saying it would not recognise the poll.
A total of 29 candidates are to run in elections for the presidencies of the three semi-autonomous Indian Ocean islands of Grande Comore, Anjouan and Moheli.
The Comoros, which has seen 19 coups attempts since its 1975 independence, had been due to elect their individual presidents on Sunday.
"[President of Anjouan] Mohamed Bacar's supporters have opened the polling stations this morning," Ahmed Mohamed Allaoui, an election official, said from the town of Mutsamudu on Anjouan on Sunday.
"The island's election commission has photocopied black and white ballot papers, but the voters are refusing to vote. It's a pretend election."
Turnout on Andoujan was low during the unofficial polls, reports said.
The African Union (AU), which has a security mission in the Comoros so the polls run smoothly, has said it and the international community would not recognise the results of the Anjouan election if officials there ignored the delay order.
EU appeal
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 Ahmed Abdullah Mohamed Sambi, the Comoros' federal president.
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.
'Intolerable'
An envoy from the AU, 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 islands were annexed by France in 1904.