Footage taken by Tracey Simpson-Laing shows water pouring from beneath a tile in the station floor.
She said the station was flooded "from inside and out".
Heavy rain battered the city for around an hour at 5pm.
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The Met Office had issued a thunderstorm notice, warning that flooding was possible.
There was also flash flooding in Central Arcade.
Flamingos Coffee House, based in the arcade, tweeted: "Flash flooding hit us today at Flamingos- no serious damage but with an early start tomorrow to finish the clean up, we will be open as usual at 10am. Sorry we had to close this afternoon!! Just what we needed..."
Pictures were also posted to Facebook of flooding in the car park of Victoria Leeds.
Light showers are forecast for much of tomorrow afternoon.
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