Search-and-rescue efforts for 121 people missing had been suspended in preparation for the controlled demolition but will resume again, officials said on Sunday.
Residents in nearby buildings were told they do not need to evacuate but should stay indoors and turn off air conditioning due to dust, the county mayor said.
Ahead of the demolition, a crowd gathered along a nearby street to witness the event. Several people stood quietly with candles in vigil for those who were lost and the still missing.
The move to demolish the partially collapsed building was considered urgent because of the approach of Elsa, North Miami Beach City Manager Arthur Sorey said.
As of Sunday afternoon, Elsa was off the coast of Cuba with winds of 60 miles per hour (95 kph). After moving across Cuba later on Sunday and Monday, the storm was forecast to approach western Florida on Tuesday or Wednesday.