A 1-1 draw at home to Gabon in Bujumbura on Saturday ensured second place for Burundi in their group, setting off wild celebrations in the small east African country.
"It's amazing, I cannot find the words. I need to take myself away for a moment to try and properly realise what has happened because I have not really been able to do that yet," Berahino told Radio France International.
"It was crazy from start to finish, from the moment we went to the stadium until when we left. It's been totally unimaginable.
"I can only thank God for the fact that I have always wanted to represent a team at a big tournament, I can do with my homeland. It's perfect. It's a dream come true."
Now 25, Berahino fled the country aged 10 along with his mother after his father was killed in civil strife.
He was given refugee status in United Kingdom, where he began his career at West Bromwich Albion, played for England at under-21 level and was even called up to the senior squad.
His early promise was, however, blighted by disciplinary problems.
Last year, FIFA approved a request to change his international allegiance to the country of his birth and Berahino has been a regular in the 2019 Nations Cup qualifying campaign since scoring on debut against Gabon away in September.
His addition to a team without much experience was a boost as was the decision to expand the size of the Nations Cup field to 24 teams, allowing two from each four-team qualifying group to progress to the finals.
"That decision made us more motivated and it gave us more of the determination we needed to get our name on this Africa Cup of Nations," he said.
"It wasn't a simple task but we are a very united squad. We were successful thanks to our unity.
"Being the first generation from Burundi to qualify for the Cup Nations is something we will always have. I am so happy for all those who have been involved in this adventure."