Stirling Moss, the era's most admired driver, was the dominant early force. He set the fastest lap of the race — a time of 1 minute 38.9 seconds recorded on lap 37 — and ran near the front in his Cooper until a suspension failure on lap 40 forced his retirement. With his own car out, Moss took over the Cooper-Climax of Maurice Trintignant and continued in the race. Under the shared-drive regulations then in force, neither man was eligible for the full allocation of championship points; they were classified as a joint third and received 1.5 points each.
Jack Brabham, the reigning World Champion and the man who had taken the title the previous season also in a Cooper, retired on lap 42 with gearbox failure. The cascade of retirements among the frontrunners opened the door for McLaren, who had been working his way through the field from an unimpressive grid position. He took the victory with Cliff Allison second in a Ferrari, a result that reflected Ferrari's continued competitive presence even as the British rear-engined constructors began to dominate.
The race introduced the Lotus 18 to championship competition. Innes Ireland qualified the new Colin Chapman design in second position on the grid, demonstrating the potential of the rear-engined Lotus immediately on its debut. The Lotus 18 would go on to win several championship races during the 1960 season, confirming that Chapman's new car would be a major factor in the championship.
Several aspects of the 1960 Argentine Grand Prix gave it an elegiac quality. For Argentine national hero José Froilán González — the driver who had delivered Ferrari its first World Championship race victory at Silverstone in 1951 — the Buenos Aires race was his final Grand Prix appearance before his home crowd. Harry Schell, the experienced Franco-American privateer and longtime fixture of the Formula One grid, also made his last championship appearance in Argentina; he was subsequently killed during practice for a non-championship race at Silverstone.
Venezuelan driver Ettore Chimeri made the only Grand Prix appearance of his career at Buenos Aires. He was killed in a racing accident at Havana two weeks after the Argentine race, leaving the Argentine Grand Prix as the sole record of his championship participation.
Three additional local Argentine drivers — Pedro Llano, Nasif Estéfano, and Alberto Rodríguez Larreta — participated in practice and qualifying sessions but did not start the race itself, sharing cars among themselves. Antonio Creus and Roberto Bonomi each made their only World Championship appearances in this race.
The 1960 race marked the end of the first era of Argentine Grand Prix racing within the World Championship. The country did not host another round of the championship for twelve years; when Argentina returned to the calendar in 1972, it was on a different layout of the Galvez circuit. The 1960 event was therefore the definitive farewell to that particular circuit configuration.
McLaren's victory from the back of the grid established him as a race winner at an age when many contemporaries had yet to score their first championship point. He would go on to found the constructor that bears his name, making the 1960 Argentine win the first milestone in one of the sport's most consequential careers.