Tremors were felt at about 4.08 pm in several areas of the state including Cuttack, Bhubaneswar, Balasore, Bhadrak and Jajpur, Dhenkanal, Nayagarh, Paradip, Rourkela and Berahmpur.
People rushed out of their houses and offices in Bhubaneswar and Cuttack as tremors were felt at least twice in the afternoon.
However, there has been no loss of life and damage to property.
"I felt back-to-back tremors within seconds. I immediately came out after I realised that the building had started shaking," said Hrudananda Sahu, a resident of Bhubaneswar.
Director of the Regional Meteorological Centre in Bhubaneswar Sarat Sahu said, "The earthquake has been felt in several parts of Odisha, West Bengal and Assam. In Odisha the tremors have been felt mainly in coastal parts as the seismic wave have travelled through the ocean."
According to National Centre for Seismology, a unit of Ministry of Earth Sciences, the epicenter of the quake was in Myanmar and it occurred at 4:04 PM at a depth of 58 kms.
Tremors were also felt in Bihar, West Bengal and northeastern states in India, spreading panic among people.
There were no immediate reports of loss to life or property.
The northeast is one of the most earthquake prone regions in the world.
There were no immediate reports of injury or damage. No tsunami warning was issued, Xinhua reported.
The temblor has an epicentre at a latitude of 42.6 degrees north and a longitude of 142.0 degrees east at a depth of 30 km.
The quake logged lower 5 in parts of Hokkaido prefecture on the Japanese seismic intensity scale which peaks at 7.
The US Geological Survey put the hypocentre of the tremor at a depth of 18 km and the epicentre at a point 30.5 km north of the city of Donggala, reports Efe news.
At least 557 people died and almost 400,000 were displaced by four earthquakes with intensities varying between magnitude-6.3 and 6.9 which struck Lombok island between July 29 and August 19.
Indonesia's deadliest ever tremor struck Sumatra island in 2014 and triggered a tsunami that killed nearly 280,000 people in around a dozen countries along the Indian Ocean, with the biggest number of casualties registered in Indonesia.
Indonesia sits on the Pacific Ring of Fire, an area of great seismic and volcanic activities, where some 7,000 earthquakes, mostly moderate, are recorded each year.
People rushed out of their houses and offices in panic. The earthquake was felt around 3.21 p.m. and lasted for 15-20 seconds.
According to Met department officials, the epicentre of the earthquake was near Jodhpur. Tremors were felt mostly in Jodhpur, Pali, Ajmer, Pushkar and Deedwana among other places.
"I felt dizzy and rushed out of my building in the open," said K.C. Joshi, an executive working in a private company. His office is located on the third floor in Jodhpur.
Pushkar resident M.C. Sharma said he saw his window panes shaking for a few seconds.
The Met department official said here: "An earthquake measuring 4.5 on the Richter scale occurred today at 5.44 a.m.
"The intensity of the earthquake was moderate and its coordinates were latitude 35.0 degrees north and longitude 74.4 degrees east.
"The epicentre of the tremor was in a depth of 150 km."
There were no immediate reports of casualties or any damage to property.
The quake, with a magnitude of 5.8, was then followed by a second quake of 5.6 closer to shore, The Sun reported on Friday.
The epicentre, with a depth of 12.0 km, was initially determined to be at 40.406 degrees north latitude and 126.755 degrees west longitude, Xinhua news agency reported.
The quake, with a magnitude of 5.8, was then followed by a second quake of 5.6 closer to shore, according to The Sun.
The US Geological Survey said both quakes were very shallow.
There have been no reports of injuries or damage.
Magnitude 5 quakes and above are not uncommon in seismically active California.
They are considered moderate and capable of causing considerable damage if they hit a heavily populated area directly.
In Mexico City alone, at least 30 people died and 44 buildings were reduced to rubble. Eight others were reportedly killed in the surrounding state of Mexico, 13 in the state of Puebla and 54 across the state of Morelos, the Los Angeles Times reported.
The temblor caused buildings to sway violently in Mexico City, including in the historic districts of El Centro and Roma, collapsing some of them and causing huge cracks to appear on the facades of others.
On Amsterdam Street, a normally tranquil road that rings a major park in the upscale neighbourhood of Condesa, a large apartment building crumbled into a pile of concrete and dust.
Hundreds of residents helped a team of soldiers, police officers and firemen search the rubble for survivors.
Some rescuers commandeered shopping carts from a nearby supermarket and formed a human chain to haul away rubble. Several times, a warning went up about a possible aftershock or gas leak, sending hordes of panicked people running.
A video recorded from a skyscraper showed giant plumes of dust rising in the air.
Police cordoned off entire blocks because of the fears that gas leaks could cause explosions. Flights into Mexico City were temporarily rerouted to other cities but the airport reopened after checks.
The US Geological Survey calculated the preliminary magnitude of the earthquake at 7.1. The epicenter was about 80 miles southeast of Mexico City in the state of Puebla.
It struck on the anniversary of a 1985 temblor, which killed thousands of people and devastated large parts of Mexico City -- a tragedy that President Enrique Pena Nieto had commemorated earlier in the day.
The quake with epicentre 71 km southwest North Bengkulu and depth at 10 km under seabed hit at 10.08 a.m. (local time), Xinhua news agency quoted an official with the agency as saying.
"We did not issue tsunami warning as the quake is not potential for tsunami," he said.
There were no immediate reports of casualties or damages.
Indonesia is prone to quakes as it lies on the vulnerable "Pacific Ring of Fire" zone.
Among the injured, 18 are in serious condition, according to the provincial government authorities.
Seventeen of the seriously injured have been transferred to the cities of Chengdu and Mianyan for treatment, reports Xinhua news agency.
More than 50,000 tourists, including 126 foreigners have been evacuated following the quake, which struck Jiuzhaigou county on Tuesday night at a depth of 20 km.
Jiuzhaigou is a popular tourist destination in the mountains on the eastern edge of Qinghai-Tibet Plateau.
It is part of the Aba prefecture and is known for its ethnic minority communities, mountainous landscape, and stunning scenery.
The moderately quake which occurred at around 8.30 a.m. was traced 22 km southeast of Malapatan township at a depth of 30 km, the Philippine Institute of Volcanology and Seismology (Phivolcs) said.
The intensity of the quake in Digos was five.
The Philippines sits on the Pacific Ring of Fire where continental plates collide, causing frequent quakes and volcanic activity.
According to the USGS, the earthquake occurred at 4.18 a.m. (local time), 180 km off the northeastern coast of North Korea, close to where the Punggye-ri nuclear test site is located, prompting intial speculation that Pyongyang had conducted another nuclear weapons test, reports Efe news.
Each of the five tests carried out so far in Punggye-ri generated seismic activity, similar to that associated with earthquakes.
The tremor was initially believed to be caused by a nuclear test. However, the USGS soon dismissed that possibility, confirming that the tremor was recorded at a depth of 559.1 km.
There was no information from North Korea on the extent of damages or casualties caused by the quake.
According to the United States Geological Survey (USGS), the epicenter of the quake that hit the state at 8.19 p.m. on Friday was near Ridgercrest, a city located 272 km north of Los Angeles, Xinhua news agency reported.
Thursday's quake jolted a remote area in California's Searles Valley. It produced more than 1,400 aftershocks, scientists said.