The move on high-caffeine drinks is the result of work by umbrella body Community Leisure UK and the Scottish government.
In total, 1,300 sites have introduced the restriction.
Public Health Minister Joe FitzPatrick welcomed the "leadership" they had shown in announcing the ban.
The drinks, which can also have high levels of sugar, have been linked to health problems in children.
The World Health Organisation has published research suggesting an association with headaches, sleep problems, irritability and tiredness.
Kirsty Cumming of Community Leisure UK said: "After working closely with both our members and colleagues at Scottish government, Community Leisure UK is delighted that all members in Scotland have chosen to ban the sale of energy drinks to young people under 16.
"This is an important step in protecting the health of young people across Scotland and we are proud to support the Scottish government's promise made in the Diet and Healthy Weight Delivery Plan."
Hospital shops and catering facilities have already introduced a similar restriction.
Mr FitzPatrick said: "I welcome the leadership shown by all publicly funded leisure centres in restricting the sale of energy drinks to young people.
"This builds on our recent ban of energy drinks to under 16s in all hospital retail units and NHS-run catering sites.
"The Scottish government recognises that consumption of energy drinks is a significant concern to parents, healthcare professionals and young people."
He added: "We have committed to a consultation on restricting the sale of energy drinks to under-16s later this year."