Jack Kelly, 59, received emergency treatment yards from the Kilmarnock dugout after his son had helped his team to a 1-0 win over Celtic.
Liam Kelly, 22, was told the news as he and his team-mates celebrated on the pitch and was visibly distraught before being moved away from the scene as paramedics treated his father.
The rest of the Kilmarnock players lifted the trophy oblivious to the unfolding tragedy but manager Kenny Shiels was aware of the situation and looked on in serious concern.
Kelly rushed to the nearby hospital where his father was taken but he died shortly afterwards. The rest of the players were told the news of the death as they toured the Ayrshire town on an open-top bus and Strathclyde Police later confirmed Mr Kelly had died.
Speaking before Mr Kelly's death had been confirmed, Shiels said: "I am all over the place, I really am. I have gone from so high an emotion to so low an emotion.
"The dressing room is very despondent and I don't know why the man above sends down these messages to us. We are thinking more about Liam than our triumphalism."
Goalkeeper Cammy Bell felt his heroics meant nothing in light of the tragic development.
Bell produced a man-of-the-match performance to keep out Celtic with a string of impressive saves, ensuring an 84th-minute goal from Dieter van Tornhout on his birthday was enough to claim the trophy. But the goalkeeper was close to tears as he spoke after the match.
"We have had some sad news coming in, so that has made everything very difficult for us as a team," the 25-year-old said. "I'm very close to Liam so it just puts things into perspective for me. If I could take away my performance I would, if Liam's dad was all right."
Celtic have also offered their condolences.
"Celtic Football Club would like to offer its sincere condolences to the family of Kilmarnock midfielder Liam Kelly whose father Jack tragically died following Sunday's Scottish Communities League Cup final at Hampden," a statement from Celtic read.
Celtic manager Neil Lennon expressed his shock at the news.
"All of the management team, players and backroom staff wish to offer our thoughts and prayers to Liam Kelly's family," he said. "Myself and the entire team are shocked and saddened with this news and our heartfelt condolences go to Liam."