Danish internationalist Laudrup put pen to paper with the Glasgow giants in a £2.5million deal back on July 21, 1994.
The winger went on to enjoy enormous success over the next four years and helped Walter Smith's side to complete nine-in-a-row.
He won the Scottish title three times, the League Cup once and the Scottish Cup once, and was also named SFWA Player of the Year twice.
The skilful attacker is now widely considered by supporters to be one of the greatest-ever players in the 142-year history of Rangers.
But the 45-year-old has recalled how his playing days were in freefall over in Italy where he had endured unhappy spells with Fiorentina and then AC Milan.
And he has told how his father - former Denmark star Finn - had warned him that the transfer to Scotland HAD to work out if he was to revive his career.
In an exclusive interview with SportTimes, he said: "I can remember a conversation I had with my father at the time I was going to sign for Rangers.
"He said to me: 'Brian, this is going to be the most important switch in your career. This move has got to be a success for you'.
"Up until then, I had been at Bayern Uerdingen in Germany for one year, Bayern Munich for two years, Fiorentina for one year and AC Milan for one year.
"My father told me: 'If you want to be a successful player then you can't be finding a new club every season. You need to find a club and stay there'.
"Rangers was that club. Joining Rangers turned out to be the best move of my career. I enjoyed every minute of it. It was very successful for me and very successful for my family.
"It was the best four years of my career in terms of playing and the best in terms of my private life. I was happy in Scotland on the park and my family and I were very happy off it."
This week in SportTimes Laudrup looks back on the circumstances that resulted in him agreeing to sign for Rangers 20 years ago.
He reveals how he realised it would be the correct decision just a few minutes after meeting manager Smith for talks at Cameron House Hotel.
And the Scandinavian, now a television pundit in his homeland, also tells of his distress at the off-field difficulties the 54-times Scottish champions have experienced in the last two years.