The 21-year-old, who represented Team GB at the London 2012 Olympics only to exit the competition in the round of 32, made a fast start in Zagreb winning all five of his first-round matches to go into the elimination stage ranked fourth.
This earned Davis a bye through the first knockout stage before he got the better of Poland's Janusz Majewski 15-10 and Nikolaos Kontochristopoulos from Greece by the same score to reach the quarter-finals.
Davis went on to guarantee himself a bronze medal with a 15-13 over Belarusian Siarhei Byk but he was unable to go any further after losing to the defending champion and world number two Alexey Cheremisinov.
This is by far the greatest result of Davis' career to date but with the team event to come he is eyeing up more silverware from Croatia.
"I am delighted with my performance," said Davis, who saw compatriot and three-time Olympian Richard Kruse finish sixth while Marcus Mepstead and Keith Cook placed 22nd and 43rd respectively.
"Although I was disappointed not to get the final and win, I am happy to get my first European medal. I am really looking forward to the team event and know we are capable of another medal.
"This is a nice platform on which to head to the worlds."