Ben Ryan's men built on Friday's confident 21-7 victory over Portugal with a moral-boosting 38-7 triumph over Hong Kong in Saturday's opener.
Argentina then scored two second-half tries to come within a whisker of victory in a tense final pool match but England held on to record a 14-12 win. A quarter-final clash with Australia, who finished eighth in the World Series, now beckons at Moscow's Luzhniki Stadium.
"We started okay, going two scores up and we were in good shape at half-time but then we gave away the ball too much in the second half and did not make the decisions we'd have liked. That let them back in and put us under the cosh a little at the end."
Wales remain on track to defend their title after stunning tournament favourites Fiji 19-14. Paul John's side knew they had to win at all costs to keep their title ambitions alive and, having struggled to overcome Tonga 28-19 earlier in the day, Wales were up against it from the outset.
But they refused to buckle under the abrasive Fijian barrage and Gareth Davies, Alex Webber and Lee Williams all managed to get over. Levani Botia then ensured a nerve-racking finale when he scored a converted try to bring the favourites to within touching distance but an immaculate defence held Fiji at bay.
Wales will now play New Zealand in the last eight and assistant coach Gareth Williams was quick to declare that Saturday's win was no fluke.
"We know the quality is there, but their work ethic and physicality against a top Fijian side was outstanding," Williams said. "The self-belief is there for all to see so hopefully we can carry on tomorrow where we left off."
Fiji will face South Africa, who finished top of Group B after thrashing second-placed Scotland 41-0 in the pool decider, while Kenya take on France in the remaining quarter-final games.