In one incident, a Palestinian carried out a knife attack at the Damascus Gate, the main entrance to Jerusalem's Old City, and was "neutralised", police said.
In the other, two knife-wielding Palestinian boys aged 12 to 13 attacked a security guard in a settler neighbourhood of annexed east Jerusalem, with one shot and wounded and the other arrested, police said.
The security guard was wounded and treated at the scene, police spokeswoman Luba Samri said.
While other knife and car-ramming assaults have continued in the occupied West Bank amid a weeks-long surge of violence, Tuesday's attacks were the first in Jerusalem in more than a week.
The unrest has seen 10 Israelis and at least 75 people on the Palestinian side -- one of them an Israeli Arab -- killed since October 1. Many of the Palestinians killed were alleged attackers.