Dan Norton's tries in the victories over France (28-0) and the USA (15-5) took him to 37 for the season, more than any Englishman has ever scored in a single HSBC Sevens World Series season.
But the day ended on a flat note for England as they went down 14-7 against Samoa in their final Pool B match.
Wales failed to reach the last eight after a shock 22-7 defeat against Spain, who now take on Samoa.
New Zealand face South Africa and Australia tackle Argentina in the other quarter-finals.
A quarter-final win for New Zealand would clinch their 10th series title in 13 years while England can make sure of third place in the table if they can repeat their victory over Fiji at last weekend's Glasgow Sevens.
Norton overhauled Rob Thirlby's 2003-04 record against France with Sam Edgerley (2) and Tom Mitchell adding further tries with Mitchell converting all four. Edgerley and Mitchell were also on target against the USA.
Against Samoa, England conceded first half tries to Taulagi Afamasaga and Tom Iosefo with Mat Turner scoring their consolation effort.
Wales had opened their account with a 19-12 win over Zimbabwe but were tripped up 22-7 by Spain - recognised as an emerging force in the sevens game - and were then thrashed 43-7 by Fiji.
Scotland had been pitched into a tough Pool D, going down 27-12 to Australia and 20-5 to South Africa. They rallied from 12-0 down to beat Portugal 26-12 on their final outing with Fedo, Colin Gregor, James Fleming and Byron McGuigan scoring.