Two Australian ADF flights were deployed on Thursday night and a third flight will head off on Friday.
Australia's response will be scaled up as needed.
All Australian high commission staff and locally engaged staff have been accounted for but the exact number of citizens is not yet known.
Australia to send troops to Solomon Islands amid protests
The decision came after a request from Mr Sogavare to Mr Morrison for help on Thursday.
"Our purpose here is to provide stability and security to enable the normal constitutional processes within the Solomon Islands to be able to deal with issues that have arisen," Mr Morrison told reporters in Canberra.
"It is not the Australian government’s intention in any way to intervene in the internal affairs of the Solomon Islands."
The Australian officers will carry lethal and nonlethal weapons "primarily but not exclusively for force protection purposes", Mr Morrison added.
Buildings in the capital's Chinatown region were set alight on Thursday as protests and riots spilled out onto the streets for a second day.
Local radio station ZFM recorded videos on social media of buildings on fire on Thursday afternoon, with scores of protesters surrounding the area.
Hundreds of protesters, some in support of the regional Malaita province which has refused to recognise China, gathered outside the Parliament House building on Wednesday demanding the prime minister step down.
Local media reported the protest turned violent when the large crowd forced its way on to the parliament grounds, before police responded with tear gas to disperse the protesters. A leaf hut next to the Parliament House building was set on fire.
The Solomon Islands Herald said the demonstration then moved towards Chinatown, with protesters damaging Chinese shops, before moving to the Chinese embassy building.
Police stations in Kukum and Naha were also damaged.
Authorities responded by imposing a 36-hour lockdown on Honiara residents, set to lift at 7am on Friday.
“Police will continue with high visibility security operations tonight, tomorrow and beyond to make sure that our streets remain calm and peaceful,” Royal Solomon Islands Police Force Deputy Commissioner Juanita Matanga said.
Freelance journalist Georgina Kekea said the lockdown had not stopped the protesters.
“They are now heading, I think, towards the CBD,” she told SBS News.
“The rioters, the looters, they’re still out there. They managed to burn down the school last night; a few more buildings.”
Honiara High School was lit up in flames and shots were heard in social media posts on Wednesday night.
Charlie, a student of the school, said the extent of damage was upsetting to witness.
“There is a lot of damage in the school compound, especially the most [damaged] one is the IT lab… very sad to see as a student,” he told local radio station ZFM.
“For me as a leader of the school, [this] caused me to cry last night.”
Prime Minister Sogavare said the “sad and unfortunate” events were designed to “bring a democratically-elected government down”.
He described the gathering as unlawful and condemned the protesters.
“They were intent on destroying our nation and destroying the trust that was slowly built among our people,” he said in a nationwide address.
He also vowed to crack down on those responsible for the protests, saying the government’s investigative arm was “at an advanced stage” of tracking them down.
“I honestly thought that we had gone past the darkest days in the history of our country. But today's events are a painful reminder that we have a long way to go.”
The unrest is believed to be related to a 2019 decision by the government to switch allegiances from Taiwan to China, among other issues around social inequality.
The protests tied into long-standing feelings around a lack of economic development in the Malaita province.
The province's leader Daniel Suidani was among the most vocal critics of the government’s move to recognise China, and opted to retain the province’s ties with Taiwan.
Mihai Sora, a research fellow at the Lowy Institute and former Australian diplomat in Honiara, said the protests have been sparked by an “unhelpful recipe of conflict” with a mixture of provincial rivalries, contemporary domestic political issues, and geopolitical competition over influence in the Pacific.
“This narrative of a central government out of touch and decisions that were not representative of the will of people in the provinces has been captured by a decade's long secessionist movement in Malaita, a group calling for the provinces independence from Solomon Islands,” he told SBS News.
“It’s married to this to the geopolitical significance of Solomon Islands’ switch and recognition from Taiwan to China to stir up what are very genuine grievances that people in Malaita seem to have.”
Competition between Taiwan and China over influence are being embroiled in domestic issues, he said.
“The Solomon Islands is a country that is already shouldering a very heavy burden to deal with its legacy of conflict, demographic indicators of a youth bulge, higher rates of unemployment, extensive periods of economic hardship due to the extended COVID period, community resentment around the state of emergency declared by the Prime Minister.
“It all boils together.”
The Department of Foreign Affairs' SmartTraveller issued a warning on Wednesday.
"The situation is evolving in Honiara with civil unrest. Please exercise care, remain where you are if it is safe to do so and avoid crowds," it said.
Federal cabinet's National Security Committee will meet this afternoon to discuss the continuing unrest.
Cabinet minister Michaela Cash told SBS News the Australian government was watching the situation "intently".
With additional reporting by AAP.