The Saturday morning strike on the Northern Bridge damaged the road surface and trolleybus overhead wires, according to Kyiv's mayor. The two bridges, which link the eastern and western parts of the capital, had not been hit before Thursday, marking an escalation in Russia's campaign.
Klitschko earlier warned that the attacks were "tearing Kyiv apart". The Southern Bridge has been struck repeatedly by Russian drones since the start of the week.
The five deaths reported by Al Jazeera come as both sides continue to trade strikes, though details of the fatalities were not immediately available. The Ukrainian capital has come under increasing pressure from Russian forces targeting key infrastructure.
The Northern Bridge, also known as the Pivnichnyi Bridge, is a critical transport artery. Authorities have not said how long repairs will take or whether traffic has been restricted.