Other officers are seen pursuing the attack out of shot, and police said that he was eventually shot and killed.
The Palestinian Authority identified the assailant as 25-year-old Aibraham Muhamed Halsi, from the West Bank village of As-Sawahra.
Haaretz reported that residents had seen a number of Israeli security forces entering the village to search Mr Halsi’s home.
Police said that a pipe bomb was found at the scene and that the bomb squad were called in to neutralise the threat.
Magden David Adom said that a border policeman was treated by paramedics at the scene and was later taken to Sha’are Tzedek Medical Centre in Jerusalem.
Israeli police described the incident as a “terror attack by stabbing” and said that the policeman remained conscious.
Israeli Prime Minister Benjamin Netanyahu said in a statement: “On behalf of all Israelis, I send my wishes for a fast recovery to the officer who was hurt, and praise the troops who worked quickly to neutralise the terrorist.”