A report on the Body Mass Index (BMI) of four and five-year-olds starting school in the 2012/13 year found that, while the proportion of children with a healthy weight increased slightly compared to the previous year, 14.6% of children were still too heavy.
A total of 8.8% of that age-group were overweight, while 3.4% had a body mass index which was classed as obese. More than 2% were severely obese.
A disparity between those in the richest and poorest areas of Scotland was also noted, with 81% of children in the most affluent areas at a healthy weight, compared to just 74% in the most deprived. While the disparity among overweight youngsters was less stark, children in areas with the poorest postcodes were almost twice as likely to be obese as those in the most expensive.
The Scottish Government, which has launched a series of initiatives in recent years aimed at improving the health of children, welcomed the latest statistics, describing them as a "small but encouraging move in the right direction".
But members of the Association for Nutrition and opposition MSPs said it was clear that more work was needed, with BMI readings among P1 children remaining broadly similar since recording began in 2001.
Alison Johnstone, Green MSP for the Lothian region and a former East of Scotland 800 metres and 1500 metres title holder, said the figures "remain very concerning".
"More action is needed to make decent food and active lifestyles available to every child," she added. "We must make sure that when all P1-3 pupils are offered a free school meal from next year, it's healthy, tasty food that they want to eat, and we need far more local and national spending to make our parks and play areas better."
The figures, based on height and weight measurements of 53,987 children, showed that girls were slightly more likely to be at healthy weight than boys, with results of 78.5% and 76.6% respectively.
Among health board areas, NHS Borders had the highest proportion of P1 pupils with a healthy weight, while NHS Western Isles had the lowest.
Chris Mantle, a Food and Health Development Worker at Edinburgh Community Food and a member of the Association for Nutrition, welcomed the small increase in children at healthy weight but said the report's findings suggested "much work was still needed".
"The data shows us that sadly as deprivation increases, healthy weight declines. In this, as in many other areas of health, we see that social inequalities often result in health inequalities," he said.
Mr Mantle's organisation runs an education scheme in the Leith area of the capital, traditionally an area of high deprivation, which encourages healthy eating and provides families with hundreds of pounds worth of vouchers over a year which they can spend on healthy food.
He said that the project had proved a great success, but called for more funding for community based projects "which work directly on the ground."
He added: "In particular, there is a dearth of funding aimed at pregnant women. The sooner in life beneficial and sustainable changes are made the better. If we continue to fail to make much of an impression on these issues we are simply storing up problems."
Sports Minister Shona Robison said: "This is a small but encouraging move in the right direction. A key focus of our action to cut obesity is to focus on early years, where evidence suggests the greatest impact can be made."