The ban, which came into effect in November last year, was brought in as part of efforts to improve the health of prisoners and staff working in prisons.
Linda de Caestecker, director of public health at NHS Greater Glasgow and Clyde (NHSGGC), which supports HMP Barlinnie, HMP Greenock and HMP Low Moss, said the response to it had been "fantastic".
Earlier this year, a study by researchers at the University of Stirling found the level of smoke in prisons fell by more than 80% in the week after the ban was implemented.
"Eight months on we can look back and hail this as a major public health success," Ms de Caestecker said.
"This was one of the best examples of partnership working in public health with NHSGGC, the Scottish Government and the Scottish Prisons Service working well together with very effective communication between three agencies.
"Providing a smoke-free prison environment for people to live, work and visit will undoubtedly improve health and change smoking behaviour - not only in prison but also as people return to the community."
Free vaping devices were offered to prisoners for two months after the ban came into effect, with around 68% estimated to be smokers - higher than the rate of 18% of the general public in Scotland.
Ms de Caestecker said demand for smoking cessation services was higher than health staff expected.
She said: "Our staff worked extremely well with SPS colleagues to make sure the change was as smooth as possible for all concerned.
"A lot of planning went into this and just before the ban we were running 18 stop smoking groups each week in the prisons we cover.
"There has been greater ongoing demand for stop smoking services than even we anticipated and we continue to provide significant levels of support to those in custody."
She added: "Our work continues as new prisoners arrive and need our help to manage in a smoke-free prison.
"The response from all three prisons has been fantastic and former smokers are feeling the benefit of the ban.
"I look forward to seeing the on-going health benefits from this in the years to come."