The officers had been carrying out a search operation in Al-Ram, northeast of Jerusalem, during which they uncovered an arms workshop, when they spotted the vehicle coming towards them, the spokeswoman said.
One of the border policemen, who "felt in danger", opened fire, she added.
A third Palestinian in the vehicle was arrested.
Israeli security forces have launched a major crackdown on underground arms workshops in the West Bank, closing 16 since the start of the year, a senior army officer said on Tuesday.
Most of the Palestinians killed were carrying out knife, gun or car-ramming attacks, according to Israeli authorities.
Others were shot dead during protests and clashes, while some were killed by Israeli air strikes in the Gaza Strip.