In a cagey first half Wales had to leave it until the dying stages before speedster Alex Cuthbert broke clear of Manuel Montero and raced clear from halfway to score under the posts. Rhys Jones added the extras to give Wales a 7-0 lead at halftime.
Wales dominated the first two and a half minutes of the second half but Cuthbert threw the ball back aimlessly when he had Warren Davies unmarked close to the Pumas line.
Diego Palma couldn't believe his luck and gleefully picked up the ball to run the length of the field to bring Argentina back into the game.
Wales camped back inside the Argentina 22 but were kept at bay until Nicky Griffiths injected pace into an attack before the impressive Rhys Jones put his captain clear and Groves did the rest.
Groves was a happy man once the final hooter went. "It was a really good effort by the boys, fair play to the Argies - they kept us out for very long periods. The character of the boys is tremendous, we were disappointed last week in Hong Kong but I think we really stepped up to the plate and deservedly won the final.
"I think the key in the final was keeping hold of the ball. We know Argentina are very dangerous on the counter attack. It's good to get over the whitewash but the hard work was done by the boys in the middle which created the space for me."
Wales were earlier knocked out of the main draw but pushed England all the way before bowing out 17-12.
England led 14-7 at the break with Cuthbert scoring Wales's try in the first half. Groves stunned England with a try in the early stages of the second half to bring Wales within two points. With Wales pushing the favourites all the way, it was left to veteran Ben Gollins to knock over a penalty to edge his side home.
The defeat put Wales into the Plate section where they face Fiji in the semi-finals. Wales showed fortitude and character to record a famous 22-21 win with first half tries by Rhodri Gomer Davies and Rhys Jones along with a second half brace by Cuthbert.