Alex Salmond called for the "offensive" decision to be reversed as he revealed a further three teachers also face having their visas turned down.
He branded the situation "extraordinary" as he accused the agency of applying rules drawn up for quite different purposes.
The teachers from Tian Jin were due to come to Scotland for a second year under a teaching programme run by the Confucius Institute backed by Strathclyde University and the Scottish Government. The programme aims to strengthen educational cooperation between China and Scotland and promote the development of Chinese language education.
The First Minister, who is visiting China at the moment, is understood to have written to Home Secretary Theresa May to express dismay that the teachers in question were denied visas to return to Scotland.
He told BBC Radio Scotland: "It is an absolutely extraordinary situation you couldn't make it up. It is deeply offensive to the Chinese and it is a decision that has to be reversed as soon as possible."
Two teachers had their visas turned down and it is likely another three will be refused, he said.
"It amounts to sabotage of a programme that everyone thinks is absolutely fantastic and doing great work in Scotland.
"Basically what the Border Agency are applying is a rule which was drawn up last year for something entirely different. It wasn't designed for this at all.
"It's pretty obvious that somebody doing their second year is going to have certain skills which you wouldn't get in your first year and the Confuscious Institutes are well agreed that actually a second-year teacher is even more invaluable than a first-year teacher."
Describing the Border Agency as "deeply dysfunctional", he said: "I'm accusing them of sabotaging a Scottish Government programme.
"The importance of this is quite fundamental. Our argument is that to have a successful economic relationship with China you must have it underpinned by a successful cultural relationship. That is the view of the Chinese, that is the view of the Scottish Government. If you undermine one, you undermine the other."
A Home Office spokeswoman said: "We have received the letter from the Scottish Government and we have been in contact with them over these cases. We are taking this matter seriously and will be looking into the decisions."
During his trade mission to China, Mr Salmond has held a meeting with state councillor Yang Jiechi who holds China's most senior foreign affairs role.
At the meeting in Beijing, the First Minister presented Mr Yang with a Mandarin version of the Scottish Government's China strategy, outlining "the four guiding principles of Scotland's engagement with the country".
The pair also discussed how Scotland and China could forge a deeper friendship, ministers said.
Mr Salmond said: "My fourth visit to China in five years reflects the importance of the trade and investment relationship between Scotland and this country, as well as the enduring friendship between our cultures that has seen collaborations and exchanges in a range of fields over recent years.
"My meeting with State Councillor Yang was extremely beneficial, and I was taken with how well informed he was about Scotland and the business and cultural links that already exist between our two countries. He exuded friendship towards the Scottish people, having been a visitor to Scotland during his career as a young diplomat."