French just edged into the top eight in the individual event for a place in Rio, with Samantha Murray 10th and Freyja Prentice 13th.
Their combined scores helped them finish ahead of Italy and Germany.
Olympic champion Laura Asadauskaite claimed the women's European gold.
Asadauskaite, who started the run-shoot a combined 59 seconds behind eventual silver medallist Elodie Clouvel of France, had already secured her place in Rio by winning the World Cup final in June.
It meant ninth-placed Alice Sotero of Italy was also rewarded with an Olympic berth in a race which saw Germany's Lena Schoneborn take bronze at the University of Bath.
French was delighted by her career-best result which secured Britain their third Olympic place - following  Joe Choong's seventh place in the men's event on Saturday and Murray's fifth at the 2015 World Championships.
"It's amazing, I never thought I would be able to do it," French told BBC Sport.
"I thought I was pretty average all day to be honest, but held it together in the run-shoot at the end and now I just have to stay strong next year."
Murray - the silver medallist at London 2012 and 2014 World champion - was expected to challenge for her first European title.
She ranked second after the swimming and fourth after the fencing stages, but dropped to ninth after the showjumping round and struggling with her shooting in the final event.
"There was a lot of pressure," Murray admitted to BBC Sport.
"I've had a difficult year with some injuries and some personal things going on so hopefully next season will be a fresh start. I'm looking forward to a clean slate."
World junior champion and 2014 Youth Olympic silver medallist Francesca Summers finished 29th in her senior international debut.

Permission is being sought for three single turbines over 200m (655ft) high at Graythorp Industrial Estate, Brenda Road West Industrial Estate and Tofts Road West.
The Department of Energy and Climate Change said it believed these would be the highest in England and Wales.
Hartlepool Borough Council has recorded about 40 objections to the plans.
More than 300 consultation responses have been received from individuals, councils and other organisations.
The scheme developer is Hartlepool renewable energy company Seneca Global Energy Ltd.
Ward Hadaway solicitor Mark Whitehead, who is acting for the company, said single, taller turbines could generate the same volume of energy as three or four typical turbines used in other schemes.
"The increase in wind speeds at height allow the volume of renewable energy generation from each turbine to be maximised, whilst having the benefit of reducing the number of turbines used," he said.
"This is intended to deliver high levels of renewable power whilst minimising the visual impact on the surrounding area."
Of those replying to the consultation to date, just over one in 10 object on grounds such as reduction of property value, obstruction of view and effect on wildlife.
Further discussions are taking place about the location of turbines planned for Tofts Road West and Graythorp Industrial Estate.
The Ramblers Association and Network Rail pointed out their current proposed site left them within "topple distance" of the nearby railway line and buildings.
The application for three turbines with a maximum tip height - measured from the ground to the top of a turbine blade pointing upwards - of 206m (677ft) is due to be considered next year.
By comparison, turbines at Northumberland wind farms such as Lynemouth and Middlemoor are about 120-125m (about 400ft) high.

Pauline King, who was in her 80s, died of injuries from a sustained attack at her home in The Avenues in Norwich. She was found on Sunday afternoon.
Alexander Kerry, 23, of Kinghorn Road, Norwich, has been charged with murder and one count of sexual assault.
He will return to Norwich Crown Court for a plea and case management hearing on 22 May.
British pentathletes claimed women's team gold as Kate French secured Olympic qualification on the final day of the European Championships in Bath.
[NEXT_CONCEPT]
Plans for what could be the highest wind turbines in England and Wales have been submitted in Hartlepool, Teesside.
[NEXT_CONCEPT]
A man accused of the murder and sexual assault of an elderly woman in Norfolk has been remanded in custody.