Two other Hamas gunmen who tried to escape the car they were all travelling in were killed by a drone, it said, adding that weapons, explosives and large sums of cash were found in the vehicle.
Hamas confirmed the death of all three men, who it said were members of its Al-Qassam Brigades armed wing.
In the village of Zababdeh, just outside Jenin, a burnt-out car riddled with bullet holes stood against a wall where the vehicle came to a halt after being pursued by an Israeli special forces unit, residents said.
Villager Saif Ghannam, 25, said one of the two other men who escaped from the vehicle was killed just outside his house by a small drone strike that shattered the windows, while a second man was killed a short distance away.
Ghannam said Israeli forces had removed the bodies but large pools of blood lay on the ground where he said the men were killed.
The incident occurred as Israeli forces kept up a large-scale operation involving hundreds of troops and police that was launched in the early hours of Wednesday morning in Jenin and Tulkarm, another volatile city in the northern West Bank, as well as the Jordan Valley.
Israeli armoured personnel carriers backed by helicopters and drones pushed into Jenin and Tulkarm on Friday while armoured bulldozers ploughed up roads to destroy roadside bombs planted by the militant groups.
The escalation in hostilities in the West Bank takes place as fighting between Israeli forces and Hamas militants still rages in the Gaza Strip nearly 11 months since it began, and clashes with the Iranian-backed Hezbollah movement in the Israel-Lebanon border area have intensified.
In the first two days of the West Bank operation, at least 17 Palestinians were killed, including the local commander of the Iranian-backed Islamic Jihad forces in Tulkarm.
Since the Hamas attack on Israel last October that triggered the Gaza war, more than 660 Palestinians - combatants and civilians - have been killed in the West Bank, according to Palestinian tallies, some by Israeli troops and some by Jewish settlers who have carried out frequent attacks on West Bank Palestinian communities.
Israel says Iran provides weapons and support to militant factions in the West Bank - under Israeli occupation since the 1967 Middle East war - and the military has as a result cranked up its operations there.
The British government said on Friday it was "deeply concerned" by Israel’s operation in the West Bank and said there was an urgent need for de-escalation.
"We recognise Israel’s need to defend itself against security threats, but we are deeply worried by the methods Israel has employed and by reports of civilian casualties and the destruction of civilian infrastructure," a Foreign Office statement said.