The dog was bleeding from its mouth and couldn't stand after the fall, witnesses said.
Five people were hurt after the crash in Williamsburg, the FDNY said.
Janice Brown, 21, was killed when Ayisha Dobson's Nissan Altima spun out of control on the bridge Monday.
Brooklyn is getting a new $4.9 million footbridge from the promenade to Brooklyn Bridge Park.
The victim was struck by the vehicle near the Brooklyn-Queens Expressway in Woodside, the FDNY said.
Three lanes of the BQE were closed Monday after four massive containers fell off a tractor trailer.
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