The shooting started on Tuesday when troops surrounded a house where they later arrested a member of the Al-Aqsa Martyrs Brigades, an armed group linked to the mainstream Fatah faction, witnesses and Israeli military sources said.
An Israeli military spokesman said the soldiers opened fire after coming under attack from protesters throwing firebombs, grenades and stones injuring two soldiers.
Medics said a 14-year-old boy died after being shot in the chest and that five other people were wounded.
The incident took place near Manger Square, the site of Jesus's birth in Bethlehem.
The shooting came hours after an Israeli soldier was killed in the Gaza Strip during an earlier raid.
More than 210 Palestinians, about half of them civilians, have been killed in Israeli operations in Gaza and the West Bank since Israel intensified its incursions after the capture of a soldier near the Gaza border on June 25.