Ben Foden touched down twice and there were tries for captain Tom Wood and replacement prop Paul Doran-Jones as England recovered from a poor start to win in Montevideo.
England conceded a penalty try after 17 error-strewn minutes and had Jonny May sent to the sin-bin, but that setback finally shocked Stuart Lancaster's men into action.
Foden responded immediately with a try in the corner and Vunipola marked his first senior England start by crashing over three times in quick succession to kill off the spirited CONSUR XV. The 20-year-old's hat-trick completed a memorable weekend for the family after his older brother, the Saracens and England prop Mako, made his debut for the British and Lions on Saturday.
England head to Salta for their first Test against the Pumas on the back of a seven-try win but the performance was not without its flaws, particularly a sloppy first quarter. The CONSUR XV - a side comprised of players from Argentina, Uruguay, Chile and Brazil - should have been 6-0 up after nine minutes but Argentinian fly-half Benjamin Madero hooked two simple penalty kicks wide of the posts.
England finally paid for a series of mistakes when the CONSUR XV were awarded a penalty try and May sin-binned for a deliberate knock-on. May acted out of desperation because CONSUR had worked an overlap just yards from the England line following a scything break from centre Francisco Sansot. That was the kick up the backside England needed. Despite being down to 14 men, they hit back immediately with Foden collecting a pass from Stephen Myler and stretching over in the corner.
England tightened up their game. They threw fewer chancy off-loads and turned to their forward pack with Vunipola scoring twice in quick succession from powerful lineout drives. Vunipola then crashed through a weak tackle to complete his hat-trick after some sharp creative play from May and Kyle Eastmond, who looked promising in midfield.
CONSUR made a lively start to the second period, with scrum-half Tomas Cubelli collecting a neat off-load from Rotherham centre Juan Socino to score under the posts.
Exeter flanker Tom Johnson did his chances of starting the first Test at openside flanker no harm and his barnstorming run, followed by a charge for the line from Kearnan Myall, created Foden's second try.
Doran-Jones, who came off the bench for Henry Thomas, scored from close range but the CONSUR XV played right to the end and forced England into long periods of defence. Uruguayan replacement Diego Magno powered over for a third CONSUR try and England lost prop Alex Corbisiero to the sin-bin after he conceded two scrum penalties in quick succession.