"Terrorist stabbing operation, terrorist neutralised," police spokeswoman Luba Samri said in a statement.
The attacker tried to stab police officers but failed and then attempted to flee but police shot him in the legs and arrested him, Samri said.
The teenage girl was apparently injured by bullet fragments from shots fired in the ensuing melee, she added.
The incident took place near Jerusalem's light rail train service that has been the scene of multiple attacks in recent months.
At the same time, according to an AFP count, 138 Palestinians have been killed by Israeli forces, most while carrying out attacks.
Earlier today, Israel destroyed the homes of two Palestinians who had carried out attacks in October.
Meanwhile, a manhunt in Tel Aviv for an Israeli-Arab suspected of killing two people in a New Year's Day attack is ongoing.
Israeli media reported that the killing of an Arab-Israeli taxi driver on the same day was connected to the attacks, but police would not confirm the information, citing a gag order.
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