The German pair, who also won Winter Olympics bronze at PyeongChang 2018, finished second in Sochi but with 980 points they now have an unassailable lead at the top of the standings.
"After we started the season with my injury, we never anticipated that we could end it with a World Championship title and a win in the overall World Cup," Eggert said.
"It was a super season. For me, the overall win in the World Cup means a lot more than the World Championships as it reflects your performance over the entire season."
Russia's Aleksandr Denisev and Vladislav Antonov took their first-ever World Cup victory at the Sanki Sliding Centre while compatriots Vsevolod Kashkin and Konstantin Korshunov took third.
"I'm lost for words. I can't even begin to describe how I'm feeling. This is the biggest achievement of my career," Denisev said.
Germany's Tobias Wendl and Tobias Arlt, who had never lost a race in Sochi, finished seventh.