The pair were ninth after the technical routine and added 44.240 points to their total with Monday's free routine, progressing with 176.890 combined.
"We've got a few things to work on for tomorrow," admitted Randall.
"It wasn't one of our best efforts. We've put in a lot of hard work and we want to work towards higher marks."
Team-mate Federici added: "We want to go back home, look at the video and give our best performance tomorrow. We really want to give everything we've got."
Great Britain's last synchronised swimming finalists were Kerry Shacklock and Laila Vakil, who finished sixth at the 1992 Games in Barcelona.
Commonwealth silver medallists Randall and Federici, who finished 14th in Beijing, will perform in the duet free final on Tuesday at 15:00 BST.
They will then compete as part of the team event on Thursday and Friday.
Jenna Randall and Olivia Federici are the first British synchronised swimming duet to reach an Olympic final in 20 years after qualifying in ninth place.