Hat-trick helps finish the Reggae Boyz; ARGENTINA 5 JAMAICA 0.
But it was Ariel Ortega - the man they call the New Maradona - who was at the heart of everything impressive about Argentina in an eventful match which also saw Derby County's Darryl Powell sent off for two bookable offences.
Ortega was quite simply unstoppable, scoring twice in a performance which will have set the World Cup big guns thinking.
For while Ortega may not have the burning acceleration of Maradona, that familiar balance and style was there as he tormented a valiant Jamaican side with an impressive array of midfield skills.
Ortega required no "Hand of God" to dispose of Jamaica, however - just the twinkling feet of a football craftsman.
His first goal in the 32nd minute came from a slick pass from Juan Veron. Ortega surged forward to place his shot coolly under Jamaican goalkeeper Warren Barrett.
His second just after halime came from another smart pass, this time from Claudio Lopez. It found Ortega surging into the Jamaican penalty box and his left-foot finish was as clinical as it was inevitable.
In between those goals Powell had managed to get himself sent off for two innocuous bookable offences on Ortega and with his dismissal went Jamaica's chance of creating an upset.
Jamaica will complain they were harshly treated but Powell was clumsy and cumbersome in his challenges and Ortega is just the sort of player whose exquisite skills FIFA are seeking to protect with their refereeing crackdown.
Much of Argentina's hopes this time rest on the goacoring exploits of Batistuta.
His first goal came in the 75th minute, courtesy of an inspired pass from Ortega, the big man thundered home his right-foot shot from the edge of the area.
His second arrived seven minutes later with another clinical right-foot finish after which he crossed himself in thanks.
Or maybe it was a prayer for a hat-trick for his third duly arrived just five minutes from time after Ortega was bundled over in the penalty area by Jamaican defender Christopher Dawes.
Batistuta swept the spot kick with a flourish past forlorn keeper Warren Barrett and the demolition job was complete.
It means Jamaica head for home, having gained valuable experience on the world stage and their team can be proud of their efforts.
The Argentinians, coached by former World Cup winner Daniel Passarella and who have not failed to go beyond the first round since 1962, can begin to dream of much greater ambitions.
Scorers: Ortega (32) 1-0, (55) 2-0, Batistuta (73) 3-0, (79) 4-0, (pen 83) 5-0.
ARGENTINA: Roa, Sensini (Vivas, 25), Ayala, Zanetti, Chamot, Simeone (Pineda, 80), Almeyda, Veron, Ortega, Batistuta, Lopez (Gallardo, 75).
JAMAICA: Barrett, Goodison, Malcolm (Boyd, 62), Sinclair, Dawes, Gardner, Powell, Simpson, Whitmore (Earle, 73), Burton (Cargill, 45), Hall.
Referee: R Pedersen (Norway).
Sent off: Jamaica - Powell.
Bookings: Argentina - Chamot; Jamaica - Powell, Cargill.