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No one was injured in the mechanical failure, which occurred on Sunday evening during operations to winch the container on to the support vessel.
The crane operator was able to move the container away from the platform and supply vessel before it plunged into the sea, where it now lies suspended by a rope connected to the crane.
It has been reported that the container initially rested dangerously close to a mass of subsea pipelines which connect into the Far North Liquids and Associated Gas System pipeline.
The 54 non-essential workers were evacuated to the Brent Bravo, Charlie and Delta platforms in the area, as well as to Aberdeen, leaving 28 workers on board the Alpha platform.
Both Alpha and Bravo, which were already shut down for maintenance and decommissioning, have been depressurised as a result of the incident.
An effort is now underway to recover the container, with a support vessel en route to the Brent field.
The UK Health and Safety Executive has launched an investigation into the incident.