A former interior minister and hardliner on immigration, Mr Nehammer said the war was also being exploited by traffickers trying to operate illegal routes while the focus of world leaders is elsewhere.
At the start of a visit to Israel, Lebanon and Cyprus, he said using the World Food Programme to ease the food shortages caused by the standstill in the Black Sea was the only way to avoid a refugee crisis.
The European Union accuses Russia of deliberately engineering the food crisis, by bombarding Ukrainian grain stores and blockading the Black Sea, to increase the pressure on western countries punishing Moscow for the war.
Mr Nehammer said Austria would offer an extra 20 million euros ($20m) for the WFP but called for the EU to do more because the refugee situation at home was “no longer manageable” as it is.
The Austrian government said 31,000 asylum applications had been made this year, almost three times the number in the same period of 2021.
It cited figures claiming Austria had a greater burden per head than almost any other EU country, but described stopping points Cyprus and Lebanon as two countries with an even greater refugee burden.
About two million displaced people are thought to live in Lebanon and “many move on to Europe and Austria,” the chancellor's office said. “This could be fuelled further by looming hunger emergencies in Africa”.
Mr Nehammer called for more powers to be given to Frontex, the EU's controversial border police, to control the bloc's external frontiers, as well as for more officers to be deployed to the agency.
“While the spotlight of European politicians and the public is on the war in Ukraine, criminals and traffickers are brutally exploiting the situation,” Mr Nehammer said.
“That threatens social harmony in our country and weakens the system for actual refugees or persecuted people, such as people in Ukraine, of whom Austria has taken 80,000 so far.”
Austria has long called for the EU to take a tougher stance on migration, having for instance insisted right up until the Taliban takeover of Afghanistan that illegal migrants could be deported there.