Four fire engines and 21 firefighters tackled the blaze at the shop in Nelson Road, which has flats above it.
The fire damaged part of the ground floor restaurant and part of the external ducting from the ground to second floor level.
Video footage posted to the Brigade's Twitter account shows smoke escaping from the top of the building as crews control the fire.
A spokesman for London Fire Brigade said: "The Brigade was called at 0843 and the fire was under control at 1041.
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