A narrow defeat was followed by a resounding victory at The Belfry in 1985. Ballesteros famously drove the 311-yard 10th at the Midlands club to set the scene.
Sam Torrance sunk the winning putt and the champagne flowed as the players celebrated on the clubhouse roof as Concorde flew past.
"There wasn't a dry eye in the house," said Elliott.
When rookie Jose Maria Olazabal and Ballesteros led from the front to help Europe to their first win on U.S. soil in the 1987 match at Muirfield Village, the transformation of the event into a clash like no other was finally complete.
Read: Seve and Olazabal: Ryder Cup revolutionaries
Olazabal, who will captain Europe in Chicago this week, says he was addicted to the Ryder Cup drug from the moment he first sampled the atmosphere.
"That 1987 Ryder Cup was very special to me -- it made me realize how special the event was and I fell in love with it straight away," he told CNN.
His partnership in four balls and foursomes (where the players take a alternative shots) with Ballesteros was to bring 11 wins and two tied rubbers in 15 matches over the course of four Ryder Cup contests.
Despite their domination, it was the United States who made a mini comeback of their own.
They tied the match at The Belfry in 1989, then wrested the trophy back in the "War on the Shore" at Kiawah Island in 1991, where Bernhard Langer agonizingly missed a tricky putt to force another tie.
The U.S. also won at The Belfry in 1993, but a European team under Gallacher's captaincy took the trophy back at Oak Hill in 1995 to spark a run of six victories in the last eight contests.
It was Ballesteros' last match as a player and he was sadly past his best, losing his last day singles.
But his tearful embrace with arch rival Nick Faldo, who had beaten Curtis Strange in the key match, is symptomatic of the spirit of the Ryder Cup, where individual performances are secondary to the team effort.
Gallacher had tasted narrow defeat as a non-playing skipper in 1991 and 1993 so victory in such fashion was sweet.
"I felt I made a few mistakes in the first two matches, but feel I learned from those mistakes for the 1995 match," he said.
The two victories for the United States since 1995 have both been on home soil: the infamous "Battle of Brookline" in 1999 and at Valhalla in 2008.
Olazabal will doubtless still have the images of 1999 deep in his memory as the U.S. team poured on to the green after Justin Leonard's putt gave him victory over the Spaniard.
But Olazabal still had his own putt to halve the 17th, meaning that golf etiquette had been breached. He missed the 25-footer and the cup was heading back across the Atlantic.