Al-Aqsa Martyrs Brigades, a group linked to the Fatah faction led by Mahmoud Abbas, the Palestinian president, claimed responsibility in a statement, citing Israeli "crimes of occupation".
Emergency services described the attack as a Palestinian "terrorist attack".
Police said they had launched a manhunt for the assailant, who is believed to have been attacked by security guards.
Micky Rosenfeld, a police spokesman, said the attacker used the axe and a knife in the assault in Bat Ayin, a settlement south of Jerusalem.
"No shots were fired," he said.
The army said all roads around the settlement had been closed and Israeli television aired footage of a large group of soldiers in combat gear gathered at an intersection.
Sceptical about peace
The attack comes two days after Israel's new government, which includes ultra-nationalist ministers and is sceptical about peace negotiations, was sworn in.
Binyamin Netanyahu, the new prime minister, has said he will still seek peace with the Palestinians, though he has given few details about his vision for a final agreement.
But Avigdor Lieberman, the foreign minister, said in remarks published on Thursday it would be difficult to make progress in peace talks with Palestinians as long as armed Hamas fighters control the Gaza Strip.
"The Palestinians must first of all confront terror, take control of Gaza and demilitarise Hamas," he told Israel's Haaretz newspaper.
"Without these, it will be difficult to move forward."
Saeb Erekat, the chief Palestinian negotiator, said Lieberman's remarks were an insult to the world powers pushing for peace.