The Family Nurse Partnership programme supports first-time parents under 20, and sees nurses visit expectant mums as often as every week during pregnancy and throughout the first two years of their baby's life.
Nurses will offer help in areas such as child development, preventative health measures, parenting skills, breastfeeding, and diet.
They will also advise on education and employment.
The scheme is the brainchild of Dr David Olds, a professor of paediatrics and director of the Prevention research centre for family and child health at the University of Colorado.
Similar programmes in the US have been shown to improve the health of young mothers-to-be, help more women into work, reduce unplanned pregnancies and also lessen child neglect.
NHS Greater Glasgow and Clyde will recruit one supervisor and eight nurses to deliver the programme, which is expected to help 205 mums. It is hoped it will improve prenatal health and increase young mums' uptake of employment, resulting in fewer unintended pregnancies and helping to reduce child neglect.
Scottish Health Secretary Nicola Sturgeon said it was "fantastic news" that the scheme was being extended.She said: "The programme has been running in the U.S. for a number of years and has had some really impressive results for the parents involved and their children.
"The programme has been running in Lothian for two and a half years now and I expect that we will continue to see similar positive results reported when a second interim evaluation report on the impact of the project in Edinburgh is published later this month.
"Intervening at the earliest possible opportunity to support those in our society who are most in need is the key to improving Scotland's health.
"That's why I am delighted that even more children and their parents are now going to benefit from this programme."
Young mums in East and West Dunbartonshire as well as Ayrshire and Arran will also benefit.
NHS Ayrshire and Arran will recruit a supervisor and six nurses who will work with an expected 155 young women from February, at a cost of £1.6 million.