Hernandez was Argentina's key man all afternoon, and imposed himself on the action from the outset, pinning England back deep into their own half with a kick to touch.
Dylan Hartley completely misjudged his line-out throw and Hernandez was lurking behind the jumpers to slot over the opening score of the match.
Goode replied with a penalty but the visitors showed more creativity and attacking threat until Goode dropped a goal and added another penalty to put England on top.
After a kick-dominated opening quarter, England's first real attempt at running rugby drew the game's opening try.
Cueto's quick handling on the right wing set up Armitage to kick ahead and the ball sat up perfectly for Banahan to touch down under the posts.
Goode's reliable kicking gave England a 10-point cushion at the break but despite their dominance, they refused to shake off the Pumas, with Hernandez's boot always keeping them in contention.
Goode's unsuccessful attempt at a third drop-goal was roundly booed by the crowd who evidently wanted to see more entertaining fare, and England took the hint, with Dan Hipkiss and the ever-active Care doing their best to rouse the troops.
Unsurprisingly, when England next scored, Care was integral to the move.
He punched a hole in the right-hand side of the Pumas defence and his off-load to right wing Cueto hit the Sale player on the knee, knocking it perfectly into the path of Armitage to touch down.
Banahan came close to setting up a third try for Tom May, but his cheeky reverse pass between his legs just evaded the centre's grasp.
Argentina did nothing to threaten the England line until it was too late, and in the last minute the Pumas were caught out by an England counter-attack, when Cueto volleyed a pass 40 metres for Armitage to race through and touch down.
England: D Armitage (London Irish), M Cueto (Sale Sharks), D Hipkiss (Leicester), T May (Newcastle), M Banahan (Bath); A Goode (Brive), D Care (Harlequins); T Payne (Wasps), D Hartley (Northampton), D Wilson (Newcastle), S Borthwick (Saracens, capt), L Deacon (Leicester), J Haskell (Wasps), S Armitage (London Irish), N Easter (Harlequins). Replacements: S Thompson (Brive), J White (Leicester), B Kay (Leicester), J Crane (Leicester), P Hodgson (London Irish), S Vesty (Leicester), M Tait (Sale Sharks).
Argentina: H Agulla; G Camacho, M Avramovic, G Tiesi, F Aramburu; J M Hernandez, N Vergallo; R Roncero, A Basualdo, J Orlandi, M Carizza, P Albacete, A Galindo, J Leguizamon, J Lobbe (captain). Replacements: E Guinazu, M Ayerza, E Lozada, A Abadie, A Lalanne, S Fernandez, L Amorosino.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.