The bank's commitment is included in the new charter for small and medium enterprises by its parent company Lloyds Banking Group.
The group has set a target of supporting 5,000 new exporters across the UK in 2016 and 25,000 by 2020.
The charter also pledges to boost lending to small businesses by Â£1bn.
Graham Blair, from the Bank of Scotland, said: "The future success of the Scottish economy hinges on small businesses looking to overseas markets for growth.
"Through this new pledge, we want to support a new legion of Scottish exporters, and ultimately to help the Scottish economy to prosper."
Lloyds Banking Group said it planned to invest Â£450m in digital technology and training in the next three years, including creating a new international trade portal enabling UK businesses to discover opportunities across the globe.
The group has trained more than 300 UK staff in international trade and has a network of export specialists across the country to help develop and advise on overseas trading opportunities.
UK Trade and Investment Minister Lord Price said: "This government has committed to having 100,000 more UK companies exporting by 2020 and this pledge by Lloyds is a welcome contribution towards this target.
"It's great to see the banks doing their bit to support small and medium-sized businesses looking to invest and create jobs."
The Bank of Scotland has pledged to help Scottish businesses become first-time exporters to boost the nation's economy.