The group did not immediately accuse anyone of carrying out the strike, although it occurred shortly after an Israeli attack killed a top Islamic Jihad commander in Gaza.
Gaza's Health Ministry says a man and a woman have been killed in the blast and two other people were wounded.
Israel blamed Ata for recent rocket fire into its territory and said he was preparing further attacks.
The strike led to sporadic rocket fire from the Gaza Strip toward Israel, Gaza residents reported, but it was unclear if it resulted in damage or injuries.
"This is an extremely risky escalation", Fawcett said. Israeli police said they closed some roads on the edge of Gaza as a precaution.
"A building in the Gaza Strip, in which the Palestinian Islamic Jihad senior leader Baha Abu Al-Ata stayed in, was attacked", Israel's military said in a statement, calling it a joint operation between the army and Israel's Shin Bet domestic security service.
The Prime Minister's Office confirmed that Israel was responsible for the airstrike which killed Al-Ata, who it described as a "senior Islamic Jihad commander" who master-minded "numerous terrorist attacks and rocket attacks against the State of Israel in recent months and who meant to immediately carry out further attacks".
An Islamic Jihad statement confirmed the death of Al-Atta, who it said had been in the midst of "heroic jihadist action".
"Our inevitable retaliation will rock the Zionist entity", the statement said, referring to Israel.
Islamic Jihad shares Hamas's ideological commitment to Israel's destruction.
Israeli analysts say that Iran, whose nuclear programme and regional activities are the focus of a USA sanctions campaign, has cultivated Islamic Jihad in Gaza as part of a network of proxy forces on Israel's borders.