Ben Ryan's men will face Samoa on Saturday morning while Scotland tackle Sevens giants Fiji after victories over France and home favourites the Arabian Gulf.
In the other quarter-finals, South Africa will face Kenya and New Zealand lock horns with Argentina.
England's new-look side were flat and disjointed in their opening 27-12 win over Tunisia and they imploded against Canada, conceding three tries in the last three minutes to give up a 24-5 lead.
They then went down 22-10 to reigning Dubai champions South Africa in their final match, but squeezed into the last eight on points difference from Canada.
Ryan said: "The journey we have taken today has certainly had twists and turns and was certainly not what we wanted.
"We could have done with some better performances but tomorrow is a Cup quarter-final and anything can happen.
"It will be a big ask for us tomorrow but we must come out and give absolutely everything."