Israel said the pre-dawn raid was aimed at Foreign Minister Mahmoud Zahhar, who it accused of complicity in the capture of an Israeli soldier two weeks ago.
Earlier, Mr Zahhar said negotiations were the only way to end the crisis.
At least 23 civilians have died in 24 hours, including nine in one family whose Gaza City home was bombed.
The leader of Hamas's military wing was among the injured in that attack.
The strike against the foreign ministry occurred at 0130 local time (2230 GMT Wednesday), when the building was likely to have been empty.
Parts of the five-storey building were destroyed, and there was widespread damage to nearby areas.
Mahmoud Zahhar leads the Palestinian foreign ministry
Israel confirmed the air strike on the Palestinian foreign ministry, and accused Mahmoud Zahhar and Hamas of direct involvement in "terrorism and violence".
"He had knowledge of the recent hostage taking, and he is part of a leadership that has orchestrated rocket attacks, countless rocket attacks, countless missile attacks against Israeli urban areas," said government spokesman Mark Regev.
But the BBC's Wyre Davies, in Gaza City, says that before the attack Mr Zahhar gave a series of media interviews suggesting both sides needed to make a deal.
Israel responded by bombing Mr Zahhar's office, offering a clear message, our correspondent says: Israel will not negotiate and insists that Cpl Gilad Shalit is released without conditions.
In Wednesday's violence, a couple and their seven children died when an Israeli air strike targeted the Gaza City home of Hamas military commander Muhammad Deif.
Hamas denied he was in the house at the time, but the BBC has learnt that he was in fact injured in the attack.
At least 14 other Palestinians died in military action as Israeli troops occupied positions in central Gaza.
Some 80 Palestinians and one Israeli have died since Cpl Gilad Shalit, 19, was seized at a border post.
Israel says its offensive is designed to free Cpl Shalit and end rocket attacks on southern Israel by Palestinians.
Last Mod: 20 Eylül 2018, 18:16