That was a 25 percent increase on the number of applications made between January and March, compared with a 15 percent rise across the entire Union.
But other countries experienced even higher jumps, with applications rising by 159 percent in the Netherlands, 123 percent in Latvia and 79 percent in Austria.
Sweden's neighbours, Finland and Denmark, also saw bigger increases of 67 percent and 66 percent respectively.
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However the Eurostat data reveals that over the summer Hungary experienced the largest number of asylum seekers relative to its population, at 3,317 per million inhabitants, compared to 1,467 in Sweden and 997 in Germany.
The figures were released as Sweden's Prime Minister Stefan Löfven held a press conference with his French counterpart in Stockholm, where the pair called on other European nations to work more closely together to help refugees and singled out Hungary's approach to migrants for criticism.
“We are shocked by the images of women at Europe's borders. Each migrant must be treated with respect and humanely. We can not accept the statements, nor the attitudes, nor the barbed wire,” Manuel Valls told reporters in the Swedish capital.
“And even less the selection (of refugees) based on religion,” he added.
Löfven echoed his comments saying: “It is not okay to say 'as long as they're not Muslims'. That is an entirely unacceptable expression.”
Valls said the continent would not be able to cope unless everyone pulled together.
“If there's not a Europe-wide response, we will not be able to handle it,” he said.
Sweden's Prime Minister Stefan Löfven greets his French counterpart Manuel Valls. Photo: Henrik Montgomery/TT
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The 28 members of the European Union remain severely divided over an EU plan for compulsory quotas which would enable the fair and equitable distribution of 120,000 refugees.
Hungary and Slovakia are reported to be blocking the plan, although several other states, mostly in the east, have also opposed the quota system.
Both France and Sweden have said that their borders will remain open for the foreseeable future, despite neighbouring countries including Germany and Finland increasing checks on people arriving on their soil from elsewhere in the European Union.
The figures released on Friday do not take into account the surge in refugees that have arrived in Scandinavia this month.
Earlier in the week separate statistics showed that 5,200 were registered in just seven days in Sweden, the highest number since June 1992 when around 5,000 were registered in a week during the Bosnian war .