Ben Ryan's side opened with a 48-0 defeat of Mexico before a narrow 28-21 win over the USA.
The USA scored all their points in six second-half minutes but a Ben Gollings try, which he converted himself, set up a Group D decider with South Africa.
Playmaker Gollings, who plays his club rugby for Toyota in Japan, then scored two tries in the 14-10 victory over South Africa which secured top spot and a last-eight match with Kenya.
Coach Ryan was happy to see his side bounce back after a poor showing on the opening day in round three in Wellington, New Zealand.
"We suffered a lot of humility on the first day last week and we worked hard this week and I'm really pleased for the players," he told www.irb.com/irbsevens.
"There's a certain amount you can do as the coach but the players have to go out there and play and they did well out there today to gain a bit of respect."
England's chances of progression have been dealt a blow after an injury to Isoa Damudamu, of the British Army and Newbury.
"Isoa fell awkwardly and I think he might have dislocated his shoulder, which would be a massive blow to us and to him," added Ryan.
"I think he's the best forward in world sevens rugby so it was a massive blow when he came off."
Championship leaders New Zealand extended their winning run to 33 matches as they cruised through to the quarter finals.
The All Blacks have won each of the three previous rounds - in Dubai, South Africa and New Zealand - and lead second-placed Fiji by 26 points in the overall standings.
They were unstoppable in recording three victories, scoring 85 points without conceding. Samoa and Argentina also progressed as group winners with unbeaten records.
Gordon Tietjens' team opened with a 19-0 victory over Wales before a 39-0 defeat of France and a 27-0 triumph over Scotland to qualify for the last eight as winners of Group A.
New Zealand will face defending champions Fiji, who finished second in Group B after a surprise 21-17 defeat to Samoa, who will play Scotland.
Argentina finished top of Group C and set up a clash with South Africa after a 19-12 victory over Kenya, who also qualify for the quarter finals after defeats of Canada and Chile.