Two fire engines and 10 firefighters from Chiswick and Acton stations tackled the blaze where a large helium cylinder was found burning in the storage facility on Essex Place.
Fire crews set up a 200m exclusion zone around the industrial area and cooled the 25l cylinder to prevent an explosion.
The storage unit and the four wheelie bins were damaged in the fire.
The London Fire Brigade said they were called at 3.01am and the fire was brought under control at 4.27am.
The cause of the fire is believed to be the careless disposal of a cigarette into one of the commercial waste bins.