Syrian state-run news Agency SANA reported that the militant had not been killed in the attacks, but that his son, Mouaz, and another person were dead and 10 other people injured.
The Britain-based Syrian Observatory for Human Rights said al-Ajouri and his wife were among the wounded.
In another air raid by Israel later on Tuesday morning, an Islamic Jihad militant was killed in the Gaza Strip while riding a motorcycle. The Israeli military said he had been about to launch a rocket from the bike.
"The [terrorist] organization's rocket launching unit posed an immediate threat," the Israeli military said in a statement.
The Palestinian Health Ministry confirmed that three people had been killed and 18 others injured in total.
The ministry said that "two Palestinians, including a woman, were killed in an Israeli attack hit a house in al-Shuja'eya neighbourhood."
Mos'ab al-Briem, the spokesman of the Islamic Jihad militant group in Gaza, called the pre-dawn killing of its military leader an "assassination" and a "declaration of war against our Palestinian people."
Gaza militants responded to the airstrikes by launching a wave of rockets into Israel. Israel's Iron Dome aerial defence system shot down about 20 of 50 rockets fired from the Gaza Strip, an IDF statement said.
Israeli aircraft targeted military sites in the Gaza Strip in response, Palestinian security sources said. The Israeli military did not immediately confirm this.
A rocket fired from Gaza slammed into a major junction in southern Israel, narrowly missing two cars. One of the two drivers was taken to hospital with shrapnel wounds, the Magen David Adom (MDA) ambulance service confirmed.
An 8-year-old Israeli girl suffered a cardiac arrest while running to a shelter in the southern Tel Aviv suburb of Holon, the MDA added.
A rocket also directly struck a house in the town of Netivot.
Schools and kindergartens were closed and shelters opened in the greater Tel Aviv area, as two rockets were intercepted over the city.
The Israeli army published images of a damaged building on Gaza City's south-eastern outskirts, saying it had been "surgically targeted."
In the press conference at the Defence Ministry, Netanyahu said the current round of violence in the Gaza Strip and Israel "may take time."
"Israel is not interested in an escalation, but we will do anything necessary to defend ourselves," he said.