More than 8,000 migrants have arrived on the island over the past three days.
PM Giorgia Meloni says Italy is being placed under “unsustainable pressure”.
Ms von der Leyen acknowledged on Sunday that the issue was “a European challenge and needs a European answer”.
At a news conference with Ms Meloni, she praised the people of Lampedusa for the support they had given to migrants, who she said had arrived on the island “simply because of its location”.
She pledged to increase support to transfer migrants out of Lampedusa as well as stepping up efforts against people smugglers who enabled dangerous and illegal journeys.
The problem could not be solved by Italy alone, Ms Meloni said. It was currently affecting “the borders, the frontier countries” but would soon involve all of the EU states, she said.
“The future of Europe depends on its ability to tackle epoch-making challenges of our time and the challenge of illegal immigration is for sure one of them.”
On Saturday the body of a newborn baby was recovered from a migrant boat.
The mother is thought to have given birth during the journey from North Africa, the Ansa news agency reports, and the death is being investigated.
The child’s body was placed in a white coffin and taken to a cemetery in Lampedusa’s Imbriacola district, according to Italian media.
Earlier this week, a five-month-old baby boy drowned during a rescue operation off the island, after a boat carrying migrants across the sea from north Africa capsized.
Ms Meloni is pushing for a European Union naval blockade to prevent boats from crossing the Mediterranean to reach Italian shores.
The Italian Red Cross said on Saturday that it was dealing with about 2,500 people at a reception centre designed for 400 arrivals.
Volunteers and staff have been providing thousands of meals all week and helping transfer new arrivals to Sicily and elsewhere.
Nearly 126,000 migrants have arrived in Italy so far this year, which is about double the number for the same period in 2022.
Ms Meloni said on Saturday that she was calling on Ms von der Leyen “to personally realize the gravity of the situation we face” and to “immediately accelerate” the implementation of an agreement with Tunisia.
The North African country has become the main departure point for African migrants attempting to reach Europe.
The EU deal, which was signed in July, is backed up by €110m ($118m; £90m) of EU cash to stop smuggling, strengthen borders and return migrants.
The surge in arrivals led to protests on Saturday by some Lampedusa residents against plans to build a new tent camp to host the migrants.
“I have two children at home. In the past years, I did not care about this issue. But now I have an instinct of protection for my children because I don’t know what will happen to Lampedusa in the future,” one of the protesters told the Reuters news agency.
“Lampedusa says stop! We don’t want tent camps. This message is for Europe and for the Italian government. Lampedusa residents are tired,” another protester said.
Jasmine Lozzelli, an activist working on Lampedusa, told the source the migrants should be sent to mainland Italy.
“It’s not a problem of numbers, it’s a problem of how you manage the reception system. If you start to do rescues in a proper way with big ships, you take them not to an island of 5,000 inhabitants, you take them to the mainland,” she said.
Source BBC News
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