England had a shaky start in the morning in their second pool match against Tunisia before winning 26-24, but by the evening they were back in fine form, running in four tries to seal the win.
Kenya were level at 7-7 before Leicester's Tom Varndell charged in up the right wing from well inside his own half to make it 12-7.
The second half was all England as they recorded a 26-7 victory and earned a quarter-final spot against Samoa, to be played at 1.50pm local time (GMT +4) on Saturday.
England: Isoa Damudamu (British Army/London Wasps), Rob Vickerman (Leeds Carnegie), Tom Varndell (Leicester Tigers), Ollie Phillips (Newcastle Falcons), Kevin Barrett (Exeter Chiefs), Ben Gollings (Gold Coast, Australia), Chris Cracknell (Exeter Chiefs). Replacements: Josh Drauniniu (Exeter Chiefs), Andy Vilk (Benetton Treviso)
Tries: Barrett, Varndell, Damudamu (2)
Conversions: Gollings (3)