The result represents a major improvement for the pairing, who finished eighth two years ago but delivered an impressive performance with a spider themed routine.
Russia's Natalia Ischenko and Svetlana Romashina claimed gold, ahead of Spain's Ona Carbonell and Andrea Fuentes and Ukraine's Daria Iushko and Kseniya Sydorenko.
The result caps a solid few days for Britain's synchro swim team, who also secured a sixth place in the team event.
Randall and Allison in medal contention
"We exceeded our targets despite the fact we've had athletes returning from injury so the results this week have had even more significance as this didn't inhibit our performance," said performance director Biz Price.
"This week has seen a another step up in our performances and we will be even more motivated to move up the rankings at the FINA World Cup next month.
"Our technical performances still need to improve as we close in on London 2012 but the free programmes have exceeded expectations and are ahead of the curve."
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