Later on Monday, eight Palestinians were injured during an Israeli army raid near the city of Nablus, according to Wafa, the official Palestinian news agency.
One of the eight was in critical condition with a gunshot to the head, the report said.
The army did not respond to requests for more information.
The attacks, which follow the fatal shooting of two Israeli settlers on Saturday, came amid a spate of violence across the occupied West Bank, leading Israel to send army reinforcements and launch raids, cordoning off the town of Hebron, and fatally shooting a Palestinian near Nablus.
Israeli Prime Minister Benjamin Netanyahu also said on Monday that Iran was behind the attacks and that Israel was considering retaliation.
The Israeli military said the shooting took place on the Route 60 motorway that connects West Bank villages with Jerusalem and Israeli settlements.
Al Aqsa Martyrs Brigade claimed responsibility for the attack, hours later.
Israel’s emergency service initially said it was treating two people for serious injuries following the incident near the Ksavir junction.
A woman was later pronounced dead after medics administered CPR at the scene.
Israeli media identified her as Sheva Nigri, 40.
The man was admitted to an intensive care unit in the Israeli city of Beersheba.
A six-year-old girl in the car was unharmed, Israel's Kan public broadcaster reported.
Israeli soldiers closed off access to Hebron and sent a large number of troops to search the area, Palestinian state news agency Wafa reported.
Israeli Channel 12 news said the Israeli army is sending more troops to bring the total number of battalions operating in the West Bank to 23 – the highest number in 15 years, it said.
The head of Hebron municipality's activities department, Mahmoud Abu Sbeih, told The National that the city's centres have been assigned to “accommodate people stuck in the city until the entry and exit points are reopened”.
Several armed groups, including Palestinian Islamic Jihad and Hamas, the militant group that controls Gaza, praised Monday's attack.
Hamas said the shooting occurred on the 54th anniversary of an arson attack on Al Aqsa Mosque.
The gun attack was “a warning message once again that Al Aqsa is a red line”, Hamas said.
“Attempting to demolish it will ignite the entire area,” Hamas spokesman Abdul Latif Al Qanoua said on Telegram.
Hours later, at least 100 Palestinians rushed to the eastern Gazan border to protest against the attack on Al Aqsa Mosque.
More than eight people were injured by tear gas and one by live ammunition during Israeli forces dispersal of the demonstrators, Palestinian state media reported.
Monday's violence happened two days after two Israeli settlers were killed in a shooting near the flashpoint Palestinian town of Huwara.
Israeli forces on Monday also searched for the perpetrator of the Huwara attack in the town of Beita near Nablus where one Palestinian was shot in the head by the troops.
His connection to the attack was unclear but a video shared by Palestinian media showed him walking down a road, seemingly unarmed, wearing a white shirt when he is suddenly struck before he collapsed.
On Monday evening, Mr Netanyahu and Defence Minister Yoav Gallant visited the site of the Hebron attack.
Mr Netanyahu blamed the “terrorist attack” on Iran, which he said “encouraged and directed” the incident. Mr Gallant accused Iran of financing the attacks.
“All options are on the table,” he said.
The West Bank has this year been the scene of some of the deadliest violence between Israel and the Palestinians in about two decades.
At least 218 Palestinians have been killed in violence linked to the Israeli-Palestinian conflict this year.
More than 30 Israelis and several foreign citizens have been killed in Palestinian attacks, according to a tally by AFP.
Three members of a British-Israeli family were killed in an ambush near an Israeli settlement in April, when gunmen attacked their car.
The violence has also led Israeli settlers to storm nearby Palestinian villages, setting cars and businesses on fire.