Hamas has been leading the protests to ease a crippling Israeli-Egyptian blockade that was imposed in 2007 in order to weaken the militant group.
Hamas leaders in Gaza have described the $15m as their first major gain of seven months of protests.
With Hamas threatening to intensify the protests and international officials warning of a looming humanitarian crisis, Mr Netanyahu allowed the cash into Gaza.
Negotiations for the financial package to Gaza began months earlier, following the Palestinian Authority's decision to cut salaries of employees in Gaza last year, an Israeli government source with knowledge of the matter told CNN.
Right-wing Israeli Education Minister Naftali Bennett, who has been critical of any arrangement with Hamas, acknowledged the communications between Liebermann and al-Thani when he said on Israel Radio on Sunday that the two were at the "height of their contacts" in July.
The transfer sparked criticism of Prime Minister Benjamin Netanyahu; opponents of the move accused him of being soft on Hamas, after a rise in tensions in recent months, which have included a number of multiple rocket barrages aimed at Israel by Gaza militants.
Benjamin Netanyahu said it was "the right step" and that he was committed to restoring quiet along the Israel-Gaza frontier and preventing a humanitarian crisis in the coastal Palestinian territory.
The sanctions against Gaza were an attempt to put economic pressure on Hamas to relinquish control of the coastal enclave to the Palestinian Authority.
It is the first instalment of $90m that Qatar has pledged to cover the salaries of thousands of unpaid Hamas employees for six months.
The money, sent by cash in three large suitcases, was brought into Gaza through Israel on Thursday by Qatar's envoy to Gaza, Mohamed Al Emadi.
Israeli Prime Minister Benjamin Netanyahu has defended allowing Qatar to transfer millions of dollars to Hamas-run Gaza despite criticism from within his own government over the move aimed at restoring calm after months of unrest. #Netanyahu #Qatar #Gaza