There was no immediate claim of responsibility for the bombing. Militants based in North Sinai have killed hundreds of soldiers and police since the army toppled Islamist president Mohamed Mursi in 2013 following mass protests against his rule.
A policeman was killed when a bomb exploded on a bridge leading to the upscale Cairo district of Zamalek, security sources said. Two of the wounded were also policemen, while the third was a civilian, they said.
The force of the explosion left a crater in the ground. The sources gave conflicting reports about whether the bomb was planted in or near a car.
On Saturday, two bombs exploded near a police station in the residential Imbaba district, causing no casualties.
Most attacks have taken place in the Sinai Peninsula, a remote but strategic region bordering Gaza, Israel and the Suez Canal, but smaller-scale bombings have become increasingly common in Cairo and other cities.
In a separate incident, the founder and leader of an Egyptian Islamist group that has targeted policemen and soldiers in greater Cairo was killed by security forces early on Sunday, security sources said.
Hamam Mohamed Attia of Ajnad Misr was shot dead during a firefight around 1 a.m. (2300 GMT) at an apartment in Giza, which is part of the Cairo conurbation, the security sources said.
There was no immediate comment from Ajnad Misr, which emerged in January 2014, on the Twitter account it often uses to claim responsibility for attacks.
Sinai Province, which changed its name from Ansar Bayt al-Maqdis after pledging allegiance to Islamic State, has claimed responsibility for much of the violence wracking the Sinai Peninsula, while Ajnad has focused on the greater Cairo area.
The last attack claimed by Ajnad Misr was a March 28 bombing near the gates of Cairo University that wounded four police officers and two civilians.
Security sources also said that unidentified gunmen attacked Sheikh Zuweid police station in North Sinai, but the militants escaped without injury to either side.