The 24-year-old clocked 101.76 seconds to finish behind Slovakia's Michal Martikan and Sideris Tasiadis of Germany in the third leg of the series.
Martikan took victory by 0.69 seconds on the Markkleeberg course to secure the 20th World Cup gold medal of his career.
"The manner in which I did it today is what I'm happy about," said Burgess.
"Third in the heats, third in the semis, third in the final. That consistency is what I had last year and it's what got me a lot of good results."
The fourth leg of the World Cup is in Ivrea, Italy from 1-3 September.
Women's K1
Ricarda Funk (GER)  103.85 (0)
Jessica Fox (AUS)  106.31 (0)
Eva Tercelj (SLO)  109.17  (0)
Men's C1
Michal Martikan  (SVK)  100.88  (2)
Sideris Tasiadis    (GER) 101.57  (2)
Adam Burgess  (GBR) 101.76  (0)
Great Britain's Adam Burgess won bronze in the men's C1 at the Canoe Slalom World Cup in Markkleeberg, Germany.