The scrum-half, who was capped 18 times for his country and played in the Calcutta Cup triumphs over England in 1964 and 1966, died at Borders General Hospital on Monday.
Better known as "Eck", he formed a formidable partnership with his stand-off Melrose club mate David Chisholm for 11 seasons at The Greenyards as well as at international level.
Melrose RFC president Mike Dalgetty said: "The club is deeply saddened by the death of Alex "Eck" Hastie. Eck was very much part of the fabric and history of our club. He partnered his great friend David Chisholm 13 times (for Scotland] and they still hold a record of remaining undefeated on ten continuous occasions. It's unlikely this will ever be repeated.
"He was an inspiration as a coach to the Melrose Colts and a wonderful family man.
"The Moose, as he was known to his friend and foes, was a warrior on the rugby field and the most modest and humble individual you could ever wish to meet off it. He will be sorely missed by the community, and personally I will miss the occasional wee blether on the High Street.
"It feels as though Eck's passing has brought to an end something that was very special for our club. His legacy will, however, remain with us forever."
The funeral of the father-of-two will take place at Melrose Parish Church on Tuesday next week at 1.15pm, followed by a burial at Wairds Cemetery in the town.
A collection will be taken at the funeral in aid of the British Heart Foundation.