The attacker, around 21 years old, was from the town of Qalqilya in the occupied West Bank and was shot dead by police, Israeli authorities said.
Video showed a man running down a road and lunging at someone through a car window while being chased.
The attack caused panic, and one witness told Israeli television he hit the assailant with his guitar, with a hole visible in the wood of his instrument.
Around a 15-minute walk from where the stabbings occurred, Biden met former Israeli president Shimon Peres.
Biden "condemned in the strongest possible terms the brutal attack which occurred in Jaffa during his meeting with president Peres, and commented that there is no justification for such acts of terror," his office said.
"He expressed his sorrow at the tragic loss of American life and offered his condolences to the family of the American citizen murdered in the attack, as well as his wishes for a full and quick recovery for the wounded."
"I ran in the opposite direction and ran into a man who was on the ground in his blood," said the woman, who gave her name as Emily.
She said she "covered him with my jacket. He was badly injured and we waited together for the ambulances to come."
The US State Department identified the dead American as Taylor Allen Force, a 29-year-old native of Texas and a US army veteran, and denounced the attack.
Violence in Israel and the Palestinian territories since October has killed 184 Palestinians and 28 Israelis.
Others were shot dead by Israeli forces during clashes or demonstrations.
Biden is due to meet separately with Israeli Prime Minister Benjamin Netanyahu and Palestinian president Mahmud Abbas today.
When meeting Peres yesterday, he spoke of an "unvarnished, complete commitment to the security of Israel. And I hope we will make some progress."
The White House has said Biden will not be pursuing any major new peace initiatives during his visit despite the wave of violence.
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