Sobotka said preparations would continue for the construction of a 370-metre (yard) barrier which would be up to four metres (13 foot) high in places, but Alfano said the feared-for crisis would not materialise and "we will show them it is money wasted".
Italian Premier Matteo Renzi has warned that closing the pass would be a "flagrant breach of European rules" and is pushing the European Commission to force Austria to hold off on a move many fear could symbolise the death of the continent's Schengen open border system.
On Thursday he described the bid to close the border as being "utterly removed from reality".
A European Commission spokesman said the body had "grave concerns about anything that can compromise our 'back to Schengen' roadmap". Its chairman Jean-Claude Juncker is expected to discuss the issue with Renzi at talks in Rome on May 5.
- 'Increasingly restrictive' -
The Vienna government is under intense domestic pressure to stem the volume of asylum seekers and other migrants arriving on its soil with the far-right surging in polls.
UN chief Ban Ki-moon hit out Thursday at what he called "increasingly restrictive" refugee policies in Europe, saying he was "alarmed by the growing xenophobia here" and elsewhere in Europe, in a speech to the Austrian parliament.
More than 350,000 people, many of them fleeing conflict and poverty in countries like Syria, Iraq and Eritrea, have reached Italy by boat from Libya since the start of 2014, as Europe battles its biggest migration crisis since World War II.
Wedged between the Italian and Balkan routes to northern Europe, Austria received 90,000 asylum requests last year, the second highest in per capita terms of any EU country.
Legislation approved Wednesday by the Austrian parliament enables the government to respond to spikes in migrant arrivals by declaring a state of emergency which provides for asylum seekers to be turned away at border points.
Some 250 police have been deployed at the Brenner Pass, a major transport link between southern and northern Europe with an average of 2,500 lorries and 15,000 cars using it every day.
Alfano said he refused a request from Sobotka to allow Austrian police to board and inspect Italian trains before they cross the border, adding that Italy would step up its own controls trains and traffic entering the pass.
Austria is preparing for a potential closure of the pass as fears grow that migrant arrivals in Italy could spike this summer as a result of the effective closure of the Balkan route into Europe.
"Seeing as there is no European plan, we have to prepare to take measures", Sobotka said Thursday, insisting Austria is "not infringing European law" with its plans to close the border.
- Fingerprinting at sea -
Rome this week unveiled plans to fingerprint migrants crossing the Mediterranean as soon as they are picked up by rescue boats.
There have long been tensions with other EU countries over migrants arriving in Italy and travelling north without being registered.
Under the EU's much-criticised Dublin Treaty, asylum claims must be processed by the first country in which refugees arrive.
Italy was warned last year by the European Commission that it must make its registration procedures more efficient.
Amid fears that warmer weather will bring another spike in migrant arrivals in coming months, Italy is pushing a plan to introduce NATO naval patrols off Libya in time for the summer -- peak season for people smuggling.
Modelled on an existing NATO operation in waters between Turkey and Greece, the plan has been backed by US President Barack Obama and is expected to be approved by alliance leaders at a summit in Warsaw in July.
Italy has also proposed an EU-funded scheme to offer African countries cash to cooperate with the fast-track repatriation of migrants deemed to have no claim to asylum in Europe.