Damu scored twice against Kenya to seal the victory
England will face Samoa in the quarter-finals of the Rugby World Cup Sevens in Dubai after a confident 26-7 victory over Kenya in their third pool match.
The sides traded two tries apiece in the first half with Kevin Barrett and Humphrey Kayange scoring for England.
In the second half England's Isoa Damu ran in two unanswered scores.
Coach Ben Ryan said: "This team is assembled for big games, this is what they do well and we didn't let them get into their stride."
England's Ben Gollings added: "Deep down we always knew we have a few more gears to go. We're a bit lackadaisical against Tunisia but tonight was a completely different story, we didn't miss one tackle."
Earlier, England had survived a superb fightback from Tunisia to seal 26-24 victory thanks largely to two tries from Tom Varndell.
The Leicester winger's scores, one a 62m effort after the restart, and tries from Chris Cracknell and Ben Gollings gave England a 26-7 lead.
But the North Africans fought back with three tries and England were left hanging on until the final whistle.
England play their quarter-final against Samoa at 1750 GMT.
Scotland narrowly missed out on the quarter-finals after losing 19-14 to South Africa.
The Springboks outscored Scotland two tries to one in the first half, with Andrew Turnbull crossing for the Scots.
Roddy Grant scored for Scotland in the second half to level the scores at 14-14 before Mpho Mbiyozo scored the winning try for South Africa.
England squad: Kevin Barrett (Exeter); Tom Biggs (Leeds); Chris Cracknell (Exeter); Isoa Damudamu (British Army/Wasps); Josh Drauniniu (Royal Navy/Exeter); Ben Gollings (Gold Coast, Australia); Ollie Phillips (captain, Newcastle); James Rodwell (Moseley); Charlie Simpson-Daniel (Gloucester); Tom Varndell (Leicester); Rob Vickerman (Leeds); Andy Vilk (Treviso).