Gregg and Kathryn Brain, and their seven-year-old son Lachlan, were urged by the government last month to exit the country voluntarily but they have now met the requirements of a visa, allowing them to remain in the UK.
Speaking on the News Hour programme, Gregg said: "We've had to put together in weeks what would ordinarily take months to achieve in terms of the paperwork you need to assemble for a Home Office application.
"To be able to have done this is a testament to the faith of a lot of people who have been working towards this goal. We really weren't sure this was possible, so it really has been a miracle."
Moving to Dingwall in 2011 on a two-year post-study visa scheme which was later withdrawn by the government, the family embarked on a high-profile campaign to stay in the Highlands.
Despite receiving support from Scottish First Minister Nicola Sturgeon and their local SNP MP Ian Blackford, Gregg and Kathryn missed the deadline last month to meet the requirements of a tier two UK visa.
The family continued their struggle and it emerged on Tuesday that Kathryn had secured a job with a Highland hotel group which fulfils the government's requirements.
Gregg went on to say: "God doesn't give us things that we can't handle, even though sometimes it may not seem that way at the outset.
"Knowing that we would be seen through this, through the bits where we felt like we couldn't manage on our own, [and] that He was with us, was a rock which kept us sane."
Asking what his message for the government was, Gregg added: "I'm hoping that the Home Office will see that treating people with honest and integrity is something that should be able to be taken for granted.
"Hopefully, having shed this light on how we've been dealt with will improve things for other people who are going through the same sort of process."
A Home Office spokesperson said: "We have always been clear with the Brain family that if a suitable job offer was received an application to remain in the UK would be considered.
"We gave them a number of extensions on an exceptional basis to allow them to try to secure a job that would allow them to meet the Immigration Rules.
"Mrs Brain was subsequently offered a job with a hotel group. This has been considered and we are satisfied that it meets the conditions for a Tier 2 visa.
"Yesterday (Tuesday 20 September), we wrote to Mrs Brain and confirmed that she and her family have been granted leave to remain in the UK."
Listen to Gregg Brain speaking with Premier's Aaron James by clicking here: