A video of the child that was posted online last week shows the boy crying with a severely inflamed testicle.
The video provoked a wave of sympathy on social media, with the launch of the Arabic-language hashtag Let_the_Rukban_child_in.
The child and his family are among tens of thousands of Syrians who are stuck around the Rukban border crossing in the northeast of the kingdom.
Jordan declared the area a "closed military zone" in June, after a suicide bombing claimed by the Islamic State group killed seven soldiers near a makeshift desert camp where more than 100,000 Syrians are stuck.
In a statement, the army thanked "all those who helped to relay the image of the sick child in order to treat him".
Jordan says it is already hosting nearly 1.4 million refugees, of whom 630,000 are registered with the United Nations.
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