Two casualties were taken to hospital in a serious condition following the blast, near the Hebron Road.
A police statement said bomb disposal experts had determined that a device exploded in the back half of the bus.
Many of the injured were on the bus but others were in a passing car. The car caught fire along with a second bus that was empty.
The head of the emergency department at Jerusalem's Hadassah Medical Center told Israeli Army Radio that some of the injuries were caused by flying metal nails and nuts.
A police spokesman told Israeli Channel 2 television that investigators were still trying to determine who had planted the bomb.
He declined to confirm reports that one of the two people in a serious condition was not carrying any identification papers and was considered a suspect.
The explosion happened in the evening rush hour sending up a huge cloud of dark smoke over the city skyline.
Emergency workers were quick to remove the charred wreckage of the two buses and a car from the road and reopen it to traffic.
For many, images from the scene here will bring back worrying memories of the bomb attacks by Palestinian militants that last took place in this city more than a decade ago.
This blast comes just over a week after Israeli Prime Minister Benjamin Netanyahu said that a decrease in the wave of attacks seen over the past six months was due to the "aggressive, responsible and systematic policy" of his government.
Palestinian militant group Hamas, which carried out a wave of bus bombings in the city in the early 2000s, praised Monday's blast, calling it "a natural reaction to Israeli crimes".
Over the past six months, 29 Israelis have been killed in a wave of stabbings, shootings or car-ramming attacks by Palestinians or Israeli Arabs.
About 200 Palestinians - mostly attackers, Israel says - have also been killed in that period.
The assailants who have been killed have been shot dead either by their victims or by security forces as they carried out attacks. Some attackers have been arrested.
Other Palestinians have been killed in clashes with Israeli troops.
At least 21 people have been wounded by a bomb explosion on a bus in south Jerusalem, Israeli police say.