A spokesperson for Iran's emergency organization, Mojtaba Khaledi, said two women were among the victims while 84 people were injured.
The first earthquake with a magnitude of 6.1 was reported at around 2 a.m. local time at a shallow depth of 10 km below the surface, according to the Seismological Center of the University of Tehran.
The epicenter was located 48 km northeast of the port city of Bandar Lengeh and 100 km west of Qeshm Island on the coast of the Persian Gulf.
It was followed by two more earthquakes with magnitudes of 6.1 and 6.3, respectively within hours, officials said.
The impact of the tremors was felt far and wide, including in neighboring Persian Gulf countries.
The five deaths were reported in the first earthquake, Bandar Lengeh Governor Foad Moradzadeh said, saying people had come out by the time two other powerful tremors struck.
The maximum damage was reported in the villages of Sayed Khosh, Konkh, Dezhgan, and Chah Ahmad, where many houses have been partially or fully damaged, locals told Anadolu Agency.
There have been more than a dozen aftershocks above 4 on the Richter scale since morning, forcing people to sit out in the scorching heat, locals said.
Rapid response teams and emergency squads of the Iranian Red Crescent Society have been present in the affected areas since early Saturday.
Interior Minister Ahmed Wahedi also visited the worst-affected areas of the province bordering Iraq, and issued instructions to local authorities to provide relief and support to the affected people, he said in a tweet.
The tremors come a week after an earthquake of magnitude 5.6 rocked the Persian Gulf province, although no casualties or major infrastructural damage was reported at that time.
The epicenter of the earthquake was between the Persian Gulf island of Kish and the port of Charak, located at 54.03 degrees longitude and 26.52 degrees latitude.
Hormozgan province, home to 14 small islands, has the largest coastline along the Persian Gulf, measuring around 1,000 km.
In November last year, the province was struck by two successive tremors with magnitudes of 6.4 and 6.3, respectively, which resulted in many casualties.
Iran is located in a seismically active zone and has seen many catastrophic earthquakes in the past.
The most devastating quake to hit the country in recent history was in 2013 when at least 34,000 people died in the city of Bam in southeastern Kerman province. The quake measured 6.7 on the Richter scale.