Brodziak's goal came on a two-on-one break with left winger Scottie Upshall. Goalie Pekka Rinne stopped a shot by Upshall, but Brodziak knocked the rebound into an open net at 7:28 of the period to break a 1-1 tie.
Tarasenko added his team-leading seventh goal of the season off a pass from left winger Robby Fabbri at 11:36.
The win was the third in a row for the Blues, their longest streak since they won their first three games to begin the season.
Right winger James Neal did not score for Nashville, ending his streak of producing a goal in at six consecutive games. It's the longest streak of his career and tied a Predators franchise record.
Goalie Jake Allen stopped 30 of 31 Nashville shots as the Blues improved their home record to 8-1-2. Allen is 6-0-2 in his last eight starts at home, and has now gone 14 consecutive starts at home dating to last year without allowing more than two goals.
The loss was the first in regulation for Rinne in eight starts this month. It was the first time he allowed more than two goals in any of those games.
The Predators tied the score 1-1 with a power-play goal by center Mike Ribeiro at 8:59 of the second period. The goal came with center Paul Stastny serving a hooking penalty and was set up when the Blues twice failed to clear the puck out of their zone.
It was only the third power-play goal allowed by the Blues at home this season in 46 opportunities and was only the second time the Predators scored on the power-play in 21 chances during their last seven games.
Center David Perron had given the Blues an early lead with his sixth goal of the season at 3:09 of the first period. Rinne stopped the initial shot by defenseman Jay Bouwmeester but could not control the rebound and Perron was able to backhand it into the net.
NOTES: After being a healthy scratch in all but one game this season, RW Ty Rattie was in the Blues lineup Saturday. D Robert Bortuzzo also returned after missing the last 10 games because of a lower-body injury. Rattie's only other appearance came Oct. 18 in Vancouver. ... The Predators sent D Anthony Bitetto to Milwaukee of the AHL for conditioning purposes Saturday. The only game Bitetto played this year was the season opener. ... The Predators return home to play Tampa Bay on Monday night while the Blues head to Boston for a Tuesday night matchup against the Bruins and their first game against former captain David Backes.