The attacks were carried out in response to a rocket fired at southern Israel on Saturday evening.
No Palestinian group has yet claimed responsibility for that attack.
The rocket struck near the border and is said to have caused no injuries.
It has been relatively calm in the border area since August, when fighting broke out over three days between Israel and Palestinian Islamic Jihad.
22-year-old shot by police
On Friday, the day before the rocket was fired at Israel, a 22-year-old Palestinian man was shot dead by an Israeli border police officer in the West Bank city of Hawara, AP reports.
In a film that has been widely circulated on social media, you can see how the 22-year-old and the policeman are in a scuffle, after which the 22-year-old is shot in the open street with four shots after the policeman has drawn his gun.
According to Israeli police, the origin of the fatal shooting must have been that the man attacked Israeli civilians and police officers with a knife.
Hawara's mayor instead believes that the police used excessive force and that the 22-year-old was already overpowered.