The bank will release two million of the £5 notes from March 2015 to coincide with the anniversary of the opening of the rail bridge.
Banks say that polymer notes are proven to be more durable than existing currency, with research finding that they stay cleaner for longer, are more difficult to counterfeit and are at least 2.5 times longer-lasting.
They will be introduced in Scotland ahead of England, where the Bank of England plans to issue them for the first time in 2016.
The Clydesdale Bank note, which is smaller than the existing currency, also celebrates the nomination of the Forth Bridge for inclusion in Unesco's World Heritage List in 2014.
It features the image of Sir William Arrol, one of Scotland's most celebrated engineers, whose company constructed the Forth Bridge.
Debbie Crosbie, executive director at Clydesdale Bank, said: "Clydesdale Bank is very proud to commemorate the Forth Bridge on our new £5 note.
"The structure is renowned across the world as an incredible feat of engineering so it was a fitting choice for a ground-breaking new banknote.
"We continue to lead the way in banknote development and, following the successful introduction of a new series of notes with 'Depth Image' holograms in 2009, we are now at the forefront in polymer currency.
"The Forth Bridge's super structure certainly lends itself to the intricate processes of banknote printing, combining security, durability and an aesthetically-striking design."
The bank said it has not made a decision about introducing plastic notes generally.
It said the new note will include a Spark Orbital security feature for the first time on UK currency.
Scottish Secretary Alistair Carmichael MP said: "I am pleased that Clydesdale Bank will commemorate the Forth Bridge on its new £5 note.
"The Forth Bridge truly reflects Scotland's position as a pioneer of engineering, construction and its recent nomination to become a Unesco World Heritage site reflects its global status.
"As this new note becomes part of everyday life in villages, towns, cities and communities across the country, it will serve as a fitting tribute to the vision of Sir William Arrol and all the people who have contributed to the building, maintenance and restoration of the bridge in its 124-year history."
In December, the Bank of England announced that it plans to issue plastic banknotes for the first time from 2016, when a new £5 note featuring Sir Winston Churchill will appear.
A £10 note featuring Jane Austen to follow around a year later will also be made from polymer rather than the cotton paper currently used.
Fiona Hyslop, Cabinet Secretary for Culture and External Affairs, welcomed the launch of the Clydesdale banknote.
She said: "Today we are celebrating two eras of Scotland's innovation and foresight.
"The introduction of this innovative new banknote featuring the iconic Forth Bridge as a symbol of Scotland's engineering heritage and ingenuity is very welcome.
"We are immensely proud of the Forth Bridge and its nomination for inclusion in Unesco's World Heritage List.
"The launch of this banknote is such a fitting way to mark this nomination and Sir William Arrol's work and I applaud the Clydesdale Bank for this gesture."
Sara Thiam, director of the Institution of Civil Engineers (ICE) Scotland, said: "Sir William Arrol is one of Scotland's most prolific civil engineers.
"Brought up in the Glasgow area, he developed methods of working and techniques which are still used today and is responsible for three of the world's most iconic bridges - the Forth Rail Bridge, the Tay Rail Bridge and London's iconic Tower Bridge.
"His legacy is littered with engineering 'firsts' so it is fitting that he is to feature on this celebratory first plastic banknote and underlines the vital contribution of civil engineers to society, past and present."