The former Saracens and Blackheath winger finished off a Dan Norton break in the first half, then added a brilliant solo effort late in the game which skipper Ben Gollings converted from the touchline.
Australia struck back after the hooter through Pat McCutcheon when Kevin Barrett was in the sin bin but James Stannard missed with his easier conversion chance.
The result puts England into a quarter final tie against Group C runners-up Samoa with Australia taking on Kenya.
England had trailed 7-0 at the break to a Bernard Foley score after Greg Barden was yellow carded but struck back through Caprice immediately after the break.
Lachie Turner and Stannard’s conversion made it 14-7 but replacement Mathew Turner went over for England before Caprice struck down the right and Gollings added the majestic extra points.
Earlier England had brushed aside minnows Sri Lanka 59-7 and Uganda 55-0, scoring 18 tries in the process.
James Rodwell scored a hat trick against Sri Lanka and Turner emulated the feat against Uganda.
Also through to the quarter finals are Wales, who went down 21-5 against South Africa and now face a daunting game against triple gold medal winners New Zealand.
Earlier they beat India 56-7 and Tonga 38-7 but have a fitness doubt over Ospreys wing Kristian Phillips, who pulled up clutching a hamstring after scoring inside 20 seconds against the hosts.
Scotland beat Guyana 26-0 and Canada 19-17 but went down 46-0 to holders New Zealand and take on the Springboks tomorrow as a result.
Tuesday 12 October
Semi-finals 9.58 and 10.20 on BBC 2
Bronze medal match 11.50 on BBC 1
Gold medal match 12.15 on BBC 1