Just fewer than half of that age group first saw e-cigarettes being used at school, according to the ASH Scotland study.
In the 15-18 age group, 48 per cent of those questioned said they had tried the devices.
More than three-quarters (78 per cent) of 13 to 18-year-olds who used them have previously tried a normal cigarette, with the majority citing "curiosity" as the reason for trying an e-cigarette.
However, nearly two-thirds (63 per cent) of the young people questioned think e-cigarettes are not "cool".
The charity is calling for laws to ban the sale of the devices to anyone under 18 and for tighter controls on their marketing.
It said there was currently no reliable evidence on the long-term impact of inhaling e-cigarette vapour, although it added this was highly unlikely to carry the levels of harm seen with tobacco smoke.
ASH Scotland chief executive Sheila Duffy said: "Our survey shows teenagers are using e-cigarettes in significant numbers, and it is particularly worrying that children as young as 13 and 14 are trying them.
"The findings underline our call for legislation to outlaw the sale of these devices to anyone under 18 and for tighter controls on their marketing.
"There is no doubt that e-cigarettes, which come in flavours such as milkshake and bubblegum, are attractive to young people. But many contain nicotine - a highly-addictive substance - and currently there is a lack of regulation of their contents and promotion.
"We also need more research into whether the use of e-cigarettes, and in particular the way they are marketed and promoted, could provide a gateway to tobacco and could "renormalise" cigarette smoking, something we must fight against as Scotland moves towards its goal of having a generation free from tobacco by 2034."
More than half (57 per cent) of those surveyed agreed that young people could be influenced to try e-cigarettes by advertising.
The survey involved 468 young people aged 13 to 18 in Scotland and was carried out between January and April this year.
A Scottish Government spokesman said electronic cigarettes "require appropriate regulation and should never be promoted to young people".
"To ensure young people are properly protected, we are committed to introducing a restriction on the age at which an e-cigarette can be purchased," he added.
"We are exploring what more can be done in Scotland now that the European Tobacco Products Directive has set out its measures for regulation and we will consider all available options to protect public health.
"We will continue to monitor emerging evidence of the impact of these devices."