Typhoon Pulasan comes days after the strongest storm to hit the megacity since 1949, Bebinca, caused extensive damage on Monday.
Pulasan made landfall on Thursday night, with a maximum wind speed of 83 kilometres (52 miles) per hour, according to the state-run Xinhua news agency.
The city evacuated 112,000 people, Xinhua said, and some ferry and train services were suspended.
Videos posted on social media Friday showed Shanghai residents wading through calf-deep water in some neighbourhoods, though no severe damage or casualties have been reported so far.
Two weather stations recorded over 300 mm of rainfall within six hours, the highest in their districts since records began, Xinhua said.
A video published by state-owned Shanghai Media Group showed police officers in high-vis coats pushing a stalled car through water in one district, while a scooter driver in a poncho attempted to cross a flooded intersection.
Around a dozen cars had broken down in that area due to the waters, according to the video.
Many of the areas that were flooded earlier in the morning were dry and cleaned up by around 11am, an AFP reporter saw.
Parts of Shanghai had upgraded their typhoon alert levels as the storm approached the city on Thursday.
The storm "is forecast to gradually weaken as it moves inland", Xinhua said, though downpours continued in the city on Friday morning.
On Monday, Bebinca felled more than 1,800 trees and left 30,000 households without electricity, with authorities evacuating more than 400,000 people across Shanghai ahead of the storm.
Scientists say climate change driven by greenhouse gas emissions is making extreme weather more frequent and intense.
China is the world's biggest emitter of greenhouse gases, though its per capita emissions pale in comparison to rival economic power the United States.
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