The humiliating loss came in humiliating fashion for one of football's most storied nations.
Argentina goalkeeper Wilfredo Caballero miskicked a clearance and Croatia defender Ante Rebic recovered the ball before sending it into the net in the 53rd minute.
Luka Modric scored another with a hooking shot in the 80th and Ivan Rakitic added the third in stoppage time.
Croatia advanced to the round of 16 with six points from two games in Group D. Argentina has only one point and will next face Nigeria on Wednesday in St Petersburg.
Argentina have won two World Cups — the last in 1986 — but have not won any major title in 25 years. They lost the World Cup final to Germany four years ago in Brazil.
Their worst loss in group play came in a 6-1 defeat against Czechoslovakia in 1958.
Croatia, 1998 World Cup semifinalists, reached the knockout stage for only the second time.