Rush, from Humboldt, Sask., and Lumsden, from Burlington, Ont., recorded a two-run time of one minute 54.41 seconds to finish just shy of the medals on the 17-corner track.
Still, it was good enough to clinch the World Cup crown.
"I have always really wanted to win an overall title, and this feels great," Rush said. "The Sochi track is awesome. Everything about it is big and impressive. It is by far the nicest facility on tour."
Rush, a runner-up last season at the world championships, won two World Cup victories and four podium finishes with brakemen Lumsden and Lascelles Brown. Rush finished 54 points ahead of Latvia's Oskars Melbardis for the title. Germany's Manuel Machata was third, 152 points back.
"It is a great way to finish off the two-man season," Lumsden said. "It truly was a team effort this year. Lascelles and I both pushed Team Rush onto the podium, and Lyndon was awesome navigating the tracks. I am very proud of what we accomplished."
Beat Hefti of Switzerland and pusher Thomas Lamparter won the race at the Sanki Sliding Centre in 1:53.76. Thomas Florschuetz of Germany was 0.30 seconds behind to finish second. Oskars Melbardis of Latvia was 0.40 back in third place.
The men's title comes a day after Canada's Kaillie Humphries and Chelsea Valois were crowned the women's overall World Cup champions.
Brown teamed up with Calgary's Chris Spring on Saturday to finish sixth. Justin Kripps of Summerland, B.C., and Neville Wright of Edmonton were 18th.
Mellisa Hollingsworth of Eckville, Alta., ended her injury-plagued season with a 14th-place finish in women's skeleton.
"It's one of the worst seasons I have had in a long time," she said. "I didn't feel prepared at all for this race, but hope when we come back next fall to put the little I did learn into action immediately and build on it.
"It has been a long season, but I'm looking forward to getting back to Canada, get refreshed and hungry for a quality summer of training."
Cassie Hawrysh of Brandon, Man., finished 16th.