Speaking at a polling station on the day of the country's general election, Mr Abiy admitted there was a problem but said the government could fix it.
The fighting, which the UN says has left five million people in need of food aid, is now in its eighth month.
More than 350,000 of them are living in famine conditions in Tigray, according to a recent UN-backed estimate.
"There is no hunger in Tigray," Mr Abiy told the BBC's Catherine Byaruhanga after he had voted. "There is a problem and the government is capable of fixing that."
Last week, the UN's humanitarian chief, Mark Lowcock, told a closed session of the Security Council that there was famine in Tigray.
He also said that starvation was being used as a weapon of war by troops from neighbouring Eritrea who are fighting alongside Ethiopian forces in Tigray. Eritrea has denied the accusation.
Mr Abiy said Ethiopia would not push the Eritreans out but was working with them to "finalise... issues peacefully".
A study released on 10 June by the UN-backed Integrated Food Security Phase Classification (IPC) initiative found that 350,000 people were living in what it described as "catastrophe/famine".
At the time, Ethiopia denied that this was the case.
A further five million people were either in "crisis" or "emergency", the study said.
The Ethiopian authorities have said that they are distributing food aid and denied reports that they are restricting access to humanitarian agencies.
'Nothing to eat'
People in Qafta Humera, an isolated district in the west of Tigray, told the BBC earlier this month that they were on the verge of starvation.
"We don't have anything to eat," one man said by phone, explaining their crops and livestock had been looted during months of war.
The conflict, which began in November last year, has forced hundreds of thousands to flee their homes and disrupted agriculture.
Ethiopia's government launched an offensive to oust the region's then ruling party, the Tigray People's Liberation Front (TPLF).
The party had had a massive fallout with Mr Abiy over his political reforms though the TPLF's capture of federal military bases in Tigray was the catalyst for the invasion.
Sudanese troops 'also in Ethiopia'
Ethiopia has allied with neighbouring Eritrea, whose troops have crossed the border and have been accused of human rights violations, including deliberately causing the lack of food - charges it denies.
In March, Mr Abiy said that the Eritrean soldiers "will withdraw" without specifying when.
At the beginning of this month his spokesperson said reports from the defence ministry indicated they had begun withdrawing.
"We are not pushing them out but we are making it peacefully, I am sure it will happen," Mr Abiy told the BBC.
"We are working with [Eritrea] to finalise our issues peacefully."
He also said that Sudanese troops were in Ethiopia, referring to the al-Fashaga triangle, which both countries claim.
Monday's general election is the first electoral test for the prime minister who came to power in 2018 as the nominee of the then-ruling coalition.
His reforming zeal, which saw the country become more open and democratic, won him supporters both inside and outside the country.
He won the Nobel Peace Prize in 2019 after ending a 20-year stalemate with Eritrea.
But the conflict in Tigray has soured his reputation.
Voting is not taking place there because of the insecurity.