Tarasenko, who made it 3-0 in the third period, has four goals and two assists in four games.
Oshie poked in a rebound of a drive by David Perron to give St. Louis a 1-0 lead midway through the first period. The power-play goal was the Blues' seventh of the season in their first 11 chances. Oshie has three goals in four games.
After Berglund made it 2-0 in the second, Tarasenko stretched the lead with 10:56 left.
St. Louis had 18 of the game's first 22 shots on goal and held Nashville without a shot for a stretch of 16:10 in the opening period. The Blues dominated play most of the night.
"Our players are starting to believe in the work part of the game," coach Ken Hitchcock said. "This was our best workmanlike effort and that's a good sign."
Nashville coach Barry Trotz acknowledged that his club was outhustled at times.
"You come into St. Louis, this is a bees' nest," Trotz said. "They get on you, they work as a group. We were not willing to play the type of game that you need to play to be competitive at this level."
Rinne had lost only once in regulation in his previous seven games against St. Louis.
The Blues improved to 4-1-1 in their last six games at home against Nashville. The Predators have not scored in the last 182 minutes, 44 seconds of regulation play in St. Louis.
NOTES: Nashville is on a seven-game road trip, tying a franchise record, that ends Feb. 5 in St. Louis. The Predators also played seven successive road games from Oct. 30 to Nov. 12, 2002. ... St. Louis D Wade Redden played his first NHL game since April 11, 2010. He was signed by the Blues on Jan. 18. ... Predators RW Patric Hornqvist has five goals and eight assists in 19 games against the Blues. ... St. Louis is a league-best 32-6-5 at home since the beginning of last season.