Couture's first goal came on a power play midway through the initial period, while his second -- which bounded off his thigh and into the net on the rebound of a Dan Boyle shot -- put the Sharks ahead by three goals 8:21 into the middle frame.
Also scoring for San Jose were Marc-Edouard Vlasic and Devin Setoguchi as the Sharks posted back-to-back wins at the end of a three-game road trip.
Nittymaki faced only one St. Louis power play in the victory.
David Backes got the Blues' sole marker on a penalty shot in the second period and Jaroslav Halak let in four goals on 30 shots.
2014: The Year in Music [PHOTOS]