A study of 250 people aged between 60 and 70 found those with a high body mass index (BMI) and big waists scored more poorly in cognitive tests.
The Alzheimer's Society said the research, in the journal Age and Ageing, added to evidence that excess body fat can affect brain function.
Lifestyle changes can help make a difference, it said.
The study looked at the relationship between fat levels and cognitive performance in adults aged 60 or over.
The participants underwent BMI - a calculation based on a ratio of weight to height - and waist circumference measurements, a scan of fat stored in the abdomen and a mental test.
Both a high BMI and high levels of abdominal fat were linked with poor cognitive performance in adults aged between 60 and 70.
In individuals aged 70 and older, high BMI, waist circumference and abdominal body fat were not associated with low cognitive performance.
The lead author of the study, Dae Hyun Yoon, said: "Our findings have important public health implications. The prevention of obesity, particularly central obesity, might be important for the prevention of cognitive decline or dementia."
A spokesperson from the UK Alzheimer's Society said: "We have all heard how a high BMI is bad for our heart but this research suggests it could also be bad for the head.
"Although we don't know whether the people in this study went on to develop dementia, these findings add to the evidence that excess body fat could impact on brain function.
"One in three people over 65 will die with dementia but there are things people can do to reduce their risk.
"Eating a balanced diet, maintaining a healthy weight, exercising regularly and getting your blood pressure and cholesterol checked can all make a difference."