Al Jazeera said on Wednesday that Abbas issued a presidential decree putting the General Administration for Crossings and Borders department under his responsibility.
The decree stipulates that the department must enjoy an independent financial, commercial and security status and be affiliated to the Palestinian presidency.
The Islamist resistance group, Hamas, which ousted Abbas' long-dominant Fatah faction in January elections on a platform of fighting corruption and Israel, decried the move as a violation of power-sharing agreements.
Officials close to Abbas said he had been under pressure from the European Union, which threatened to withdraw its monitors from the key Rafah crossing between Gaza and Egypt in response to Hamas's political rise.
Citing security concerns, Israel has also repeatedly closed the Karni commercial crossing - a lifeline for Palestinians in Gaza, from which Israel withdrew last year after 38 years of occupation.
Israel continues to control major crossings in the West Bank, another territory captured in the 1967 Middle East war and where Palestinians seek statehood.
The Hamas government said Abbas's announcement ran counter to understandings whereby control of borders would remain in the hands of the Palestinian government.
"Any attempt to reduce the authorities of the government will harm its performance and its ability to carry out its duties," said cabinet spokesman Ghazi Hamad.
In a separate development, a Jewish settler who last year shot and seriously wounded a Palestinian civilian in Gaza has been sentenced by a Bir Shiva court to two years in prison, Aljazeera.net's Khaled Amayreh reports from the West Bank.
The Palestinian victim reportedly remains incapacitated by a potentially lifelong disability.
The incident occurred in June when Daniel Finner, 39, shot and seriously wounded a Palestinian civilian at the Muwasi neighbourhood in central Gaza.
The presiding judge, Rachel Barkai, said the defendant did not think that shooting a Palestinian was of any importance and returned to his home in the settlement of Tapuah, completely confident that he wouldn’t be punished for his action.
The killing took place before the
Israel withdrawal from Gaza
Palestinian judicial officials denounced the "extremely light sentence", calling it "a tacit encouragement to murder".
Hanna Issa, director-general of the Palestinian Authority's Ministry of Justice, told Aljazeera.net that when Palestinians are wrongfully victimised or even killed by Israeli settlers or soldiers, Israeli courts even ignore the country's own criminal laws.
He said Finner should have received at least a 10-year jail sentence for deliberately and premeditatedly trying to kill the Palestinian.
Issa added that he didn't believe the defendant would complete his term in jail.
"I have no doubt that he would be released very soon, either by a presidential pardon, or because of the occurrence of a holiday or simply for good conduct."
Aljazeera.net's Khaled Amayreh in the West Bank contributed to this report