Saracens wing Noah Cato struck twice in the opening 15 minutes with Worcester's Miles Benjamin also touching down as England opened a 19-3 lead.
But they had to dig in after the interval as the Pumas forced their way back into the game before replacement full-back David Smith (Northampton Saints) and flanker Andy Saull (Saracens) scored in the final moments.
The win means England will now play-off for fifth place on Saturday against the winner of Wednesday's game between Samoa and France.
England U-19
Alex Goode (Saracens); Miles Benjamin (Worcester Warriors), Alex Tait (Newcastle Falcons) (David Smith, Northampton Saints 48), Gareth Griffiths (Bristol Rugby) (Greg King, Worcester Warriors & Hartpury College, 32), Noah Cato (Saracens); Adam Greendale (Leeds Tykes & Hymers College), Danny Pointon (Northampton Saints) (Richard Bolt, Exeter Chiefs 23-29, 42); Billy Moss (Bath Rugby) (Jack Steadman, Bristol Rugby 66), Joe Williams (Gloucester Rugby), Nathan Catt (Bath Rugby), Ben Thomas (Saracens) (Matthew Cox, Worcester Warriors 48), Gregor Gillanders (Leicester Tigers), Jon Fisher (London Irish & Wellington College), Garth Dew (Sale Sharks) (Andy Saull, Saracens, 60), Hugo Ellis (London Wasps, captain) (Tom Denton, Leeds Tykes 68)
Replacements (not used)
Alex Corbisiero (London Irish), Scott Freer (Leeds Tykes), Joe Simpson (London Wasps), Tom Bedford (Exeter Chiefs)
Argentina U-19
Belisario Agulla (captain); Nicolas Andres, Juan Pablo Socino, Juan Pablo Estelles, Diego Palma; Matias Thomas, Martin Landajo; Javier Perez, Roberton Telerizo, Juan Figallo, Anibal Panzeira Garrido, Guillermo Roan, Marcos Insua, Guido Lofiego, Lisandro Ahaulii De Chazal
Replacements
Ignacio di Santi, Jeronimo Negroto, Facundo Corimayo, Cristian Etchart, Juan Mangione, Federico Durba, Manuel Trebucq, Jorge Hill Basilio, Nicolas Sanchez, Francisco Panessi, Ignacio Poet Belmonte
