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.
British pentathletes claimed women's team gold as Kate French secured Olympic qualification on the final day of the European Championships in Bath.