The official, at a command centre used by armed groups in Gaza, said they had been "responding to crimes" by Israel - a reference to the killing on Tuesday, in disputed circumstances, of two Hamas gunmen.
Thursday's events are the latest in a series of severe flare ups over the past few months in Gaza, leading many local and worldwide officials to speculate that another large-scale Israeli offensive on the Palestinian territory could be imminent.
Military spokesman Jonathan Conricus said he had no information on the woman and child being killed, but noted a pregnant Israeli woman was taken to hospital Wednesday night after a rocket landed nearby.
Abu Mujahed, spokesman for the Popular Resistance Committee, a smaller faction in Gaza, said on his Facebook page: "Upon Egyptian and worldwide efforts, a ceasefire between Palestinian resistance began conditional on the commitment of the occupation".
The Republican Jewish Coalition is headed by Sheldon Adelson, chairman and CEO of the Las Vegas Sands Corp.
It was the third severe flare-up of violence since July and came despite attempts by United Nations officials and Egypt to secure a long-term truce between Israel and Hamas.
The flare-up, however, will distract from finding a more lasting solution to Gaza's long term economic problems, he said.
According to the IDF statement, the building served Hamas to manage internal security throughout the Gaza Strip and some of the unit's personnel were also active in the terrorist organization's military wing.
Israel said it is defending its border and accuses Hamas, a group sworn to its destruction, of using the protests as cover for attempts to breach the border fence and attack civilians and soldiers. The airstrike set off a powerful explosion and sent a huge plume of black smoke into the air, causing crowds to scream in panic.
The strike hit a building that Palestinians say housed a cultural center and other offices in the middle of the city, an AFP correspondent said. Across the border rocket warning sirens sounded nearly non-stop from sunset on Wednesday in Israeli towns and villages where residents sheltered in bunkers.
A pregnant Palestinian woman and her 18-month-old child, and a militant from the Islamist Hamas group that rules Gaza, were killed in the Israeli attacks, and at least five civilians were wounded, local medical officials said.
Public pressure in Israel to take more decisive military action is building.
In turn, Israel carried out over 140 airstrikes in the Gaza Strip. Seven people were wounded by Palestinian rockets and mortars that struck Israel.
A Hamas militant was also killed in the overnight strikes and at least 12 others injured, the health ministry said.
Palestinian president Mahmud Abbas called on the worldwide community to "immediately and urgently intervene".
He added that "if the current escalation however is not contained immediately, the situation can rapidly deteriorate with devastating consequences for all people".
Israel has killed more than 150 people since March when Palestinians began to stage protests on the border to demand an end to the blockade and the return of the Palestinians to the occupied territories.
Rocket warning sirens sounded nearly non-stop in the southern Israeli towns and border communities near Gaza from sunset on Wednesday. This comes less than 24 hours after Israeli airstrikes killed three Palestinians on Wednesday night.