A professional bobsleigher since 2002, Voyevoda won silver in the four-man bobsleigh event with teammates Philippe Egorov, Alexei Seliverstov, and Alexandre Zoubkov at the 2006 Winter Olympics in Turin. He also won a bronze in the two-man event at the 2008 FIBT World Championships in Altenberg, Germany. At the 2010 Winter Olympics in Vancouver, Voyevoda won a bronze in the two-man event. At the 2014 Winter Olympics in Sochi, Voyevoda initially won a gold medal in the two-man event and a gold medal in the four-man event. Voyevoda received the Order For Merit to the Fatherland Award 4th class with Russian President Vladimir Putin handing the state awards.