About 30 Palestinians were injured, some of them critically, the ministry said.
The Palestinian Health Ministry said the dead included four women and four children in the strikes on Gaza and Rafah.
Russians Jamal Khaswan and his wife and son were also among those killed in Israeli air strikes on Gaza, according to the Russian mission in the West Bank.
About 30 Palestinians were injured, some of them critically, the ministry said.
Militant groups said they would retaliate.
Israel’s army justified the operation by citing rocket attacks from Gaza on the Israeli border area in recent weeks.
The Israeli army said the three militants were Chalil Bahitini, a commander in the northern part of the Gaza Strip who was responsible for recent rocket attacks on Israel.
Jahed Ahnam, head of the military council and Tarek Az Aldin, coordinated the attacks in the West Bank.
UN envoy, Tor Wennesland, condemned the civilian deaths as unacceptable.
“I urge all concerned to exercise maximum restraint to avoid an escalation.
“We must be prepared for every scenario – the IDF and security forces are prepared to defend every front,” Israeli Defence Minister, Yoav Gallant, said.
The military was ordered to prepare for a possible mobilisation of reservists.
The opposition backed the Israeli government’s deployment.
The U.S. was informed about the plans, according to media reports.
Several Arab countries, including Egypt and Jordan, condemned the Israeli attacks.
Civilians in southern Israel were ordered to stay near a designated shelter until Wednesday.
Border crossings with Gaza were closed and regional rail traffic was restricted.
According to eyewitnesses, the Gaza region saw little traffic, while schools, universities and all ministries and public services were closed.
Amid fears of escalation, Managing Director of the Institute for National Security Studies (INSS), Tamir Hayman, has said.
“As far as Israel is concerned, Hamas is not the target of the operation but the main question that will determine the intensity of this conflict and its duration is whether or not Hamas will join the campaign.’’
Israeli news site Ynet reported that Israel had sent a message to Hamas that they were not targeted.
A Hamas spokesman said that the Palestinian people know how to respond to the crime of the targeted killing of Jihad members and attack the occupying power.
A Jihad spokesperson said that Israel had ignored all the initiatives of the mediators.
In August last year, Israel killed jihadi military chief, Chalid Mansur, in an air strike.
Two other jihad members were killed, including Mansur’s deputy.
At that time, there were massive rocket attacks from the Palestinian territory and further Israeli air strikes.
An Egyptian-brokered ceasefire came into effect three days later.
The latest tensions stem from the death of Khader Adnan a week ago.
The senior member of the Islamic Jihad movement died after spending almost three months on hunger strike in an Israeli jail.
The Gaza Strip is home to more than two million people living in very poor conditions, while Hamas seized power in the Palestinian territory by force in 2007.
In response, Israel tightened a blockade of the coastal area, which was supported by Egypt.
The U.S., the EU and Israel classify Hamas and Islamic Jihad as terrorist organisations.
Both groups were committed to the destruction of Israel.
Islamic Jihad, however, was seen as more radical than Hamas.
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