The confrontation Monday took place near one of the gates to Jerusalem's Old City, where the holy site at the heart of much of the recent tension is situated.
Border police officers were suspicious of a man walking with his hands in his pockets and asked him to stop and take them out, Israeli police spokeswoman Luba Samri said.
As the officers approached him, the man took out a knife and stabbed one of them in his flak jacket, Samri said. Border police responded by shooting and killing the Palestinian, she said without providing details on his identity.
Close friends of the Palestinian's family identified him as Mustafa al Khateeb, an 18-year-old from East Jerusalem who was in his last year of high school.
The police officer was unharmed in the clash, Sabri said.
The violence Monday followed a weekend of deadly clashes and an order by Israeli Prime Minister Benjamin Netanyahu for police reinforcements in Jerusalem.
The Israeli Air Force bombed two Hamas weapon manufacturing facilities in northern Gaza early Sunday in response to a rocket fired into southern Israel.
The exchange of fire suggested the Israeli-Palestinian tensions were spreading further beyond Jerusalem and the West Bank.
The Israeli strike on Gaza City caused a house to collapse, resulting in the deaths of a 3-year-old child and a 35-year-old woman who was five months pregnant, the Gaza City Fire Department said.
Israel's Iron Dome missile defense system intercepted the rocket from Gaza with no injuries or damage reported, according to the Israeli military. Another rocket launched from Gaza Sunday hit an open area in southern Israel without causing any reported injuries, it said.
Two young Palestinian men carried out knife attacks near Jerusalem's Old City on Saturday in which several police officers and other Israelis were hurt, according to police. Both of the attackers were killed by police, authorities said.
Six other Palestinians were killed in clashes with Israeli security forces over the weekend, according to the Palestinian Red Crescent. They included a 13-year-old boy who was hit by a rubber-coated bullet in the West Bank on Sunday, according to the Palestinian Health Ministry.
Hundreds of other Palestinians were wounded in the clashes in Jerusalem, the West Bank and Gaza, the Red Crescent said.
The Health Ministry says that 24 Palestinians have been killed since the beginning of October and more than 1,300 have been wounded by live and rubber coated bullets.
Four Israelis have been killed and several others wounded in knife and gun attacks by Palestinians in Jerusalem and the West Bank since October 1.
Some have suggested the violence represents the start of the third intifada, or uprising, by Palestinians. But others have dismissed that label, saying the unrest is simply the consequence of the absence of any move toward peace.
"We've tried negotiations and it didn't work. So now we will fight," one Palestinian youth in the West Bank city of Hebron told CNN as thick smoke rose from flaming tires.
Amid the continuing attacks, about 1,300 reserve border police officers have been mobilized in Jerusalem and throughout Israel, the Prime Minister's office said in a statement over the weekend.
The additional force will continue as necessary as "a primary preventive and deterrent measure," the statement said.
"We are in the midst of a wave of terrorism originating from systematic and mendacious incitement regarding the Temple Mount -- incitement by Hamas, the Palestinian Authority and the Islamic Movement in Israel," Netanyahu said.
The Temple Mount is the Old City holy site that Muslims call the Noble Sanctuary.
Palestinians have repeatedly clashed with Israeli security forces at the site in recent weeks, prompting Israel to restrict access to the site.
Palestinian leaders have suggested the Israeli government is planning to change the status quo at the site, where Jews are allowed to visit but not pray.
Netanyahu has denied the allegations and called on both sides to abstain from going to the site to avoid escalating the situation.
CNN's Michael Schwartz reported from Jerusalem, and Jethro Mullen wrote from Hong Kong. CNN's Michael Martinez, Ben Wedeman and Salma Abdelaziz contributed to this report.