After opening their Pool A campaign with a 19-10 win over Argentina on Friday, England picked up on Saturday with a 21-17 victory against Kenya courtesy of Dan Norton's hat-trick.
England's pool campaign finished with a 14-12 defeat to Samoa, despite tries by Rob Vickerman and captain Greg Barden.
England head coach Ben Ryan said: "From the Samoa game there are one or two things we need to sort out but overall I was pleased with what we did.
"We played well against Kenya to get into the quarter-finals and against Samoa we stepped up in performance and created chances."
Defeat to the unbeaten Samoans meant England finished second in Pool A and will now face a young Australian side who finished second in Group C after wins over Scotland (29-14) and France (31-10) and a 22-5 loss to Fiji.
Wales also booked their place in the quarter-finals despite winning just one of their three matches in Pool B.
The Welsh suffered a 10-0 defeat to South Africa on Friday before thumping the United States 42-7 on Saturday and concluding with a 24-7 loss to New Zealand.
Next up for Wales is a repeat of their pool match against the All Blacks in the last eight.
The other quarter-final ties see Fiji play Argentina and Samoa take on South Africa.
Scotland are out of the running for the main competition after two defeats in Group C. Despite a 14-12 win over France in their first match today, a subsequent 29-14 loss to Australia allied to yesterday's 39-5 defeat to Fiji proved their undoing.
Scotland instead drop into the Bowl competition semi-finals, where they will face the United States.
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The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
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Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
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The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.