Randall and partner Olivia Federici scored 89.170 in their free routine and, when added to their 88.1 from Monday’s technical routine, gave them an overall score of 177.270.
And 23-year-old Randall, the most successful member of the British synchronised swimming team, believes the duo are continuing to improve after bettering their 14th place from Beijing.
“In Beijing we didn’t make the final so it was great to be in there. Hopefully for the next Olympics, we will be even higher up,” said the three-time Commonwealth silver medallist.
“There are always things we can fix for our next competition, improve on and step up. It felt like a good swim and we definitely enjoyed the final.”
Russia’s world and European championship winners Natalia Ishchenko and Svetlana Romashina defended their gold medal with 197.1 after scoring a huge 98.9 for their free routine.
Spanish duo Ona Carbonell Ballestero and Andrea Fuentes Fache took silver, marginally edging China’s Huang Xuechen and Liu Ou into third.
The British duo held ninth place overnight and were able to endure their position after the free routing to post Team GB’s best placing since Barcelona 1992.
And 22-year-old Federici insists she is pleased with their performance as the duo look ahead to starting the team competition on Thursday – where their seven teammates will make their Olympic debuts.
She said: “I think you can be happy if you give everything and it’s our best performance to date. It can always be more perfect.
“The team will be the first Olympics for all of them except for Jenna and I, so it will be really exciting for them to join us as a team and feel the crowd.”