Vladimir Tarasenko and Alexander Steen had the other goals for the Blues, who have won seven of their last eight. The game also marked the fifth and final game of the season series between the Central Division rivals, with the Blues winning all five matchups.
Ryan Miller made 30 saves to improve to 6-0-1 since the Blues acquired him in a trade with Buffalo on Feb. 28. Two of those victories came against the Predators.
St. Louis is 19-0-2 against the Central this season.
Eric Nystrom scored for Nashville, which saw its three-game winning streak come to an end. The Predators have lost their last five home games, all in regulation.
Berglund scored the game's first goal at 11:32 of the opening period. Nashville failed to clear the puck from inside its blue line, and Berglund grabbed it and skated to the top of the left faceoff circle, where he beat Pekka Rinne with a wrist shot.
Rinne started for the second game in as many nights, his first back-to-back set since returning to Nashville's lineup after missing 51 games with an infection in his right hip.
Berglund struck again at 2:02 of the second. He carried the puck into Nashville's zone on the right side and pulled up at the right faceoff dot, where he beat Rinne between the pads with a wrist shot.
It was Berglund's third two-goal game of the season.
Tarasenko made it 3-0 at 7:21 of the second when he collected a rebound off of the end boards and beat Rinne to the far post to record his 21st goal of the season. He has points in five consecutive games.
Nystrom scored Nashville's lone goal at 16:50 of the third after he collected the puck following a goalmouth scramble. The goal was Nystrom's 13th of the season.
Steen scored his 30th at 8:05 of the third on a one-timer from the top of the right circle off of a pass from Jaden Schwartz. It is the first 30-goal campaign of Steen's career.
NOTES: Nashville LW Viktor Stalberg played in his 300th career regular season game. ... This is the first time the Blues have swept the season series with Nashville. ... Tarasenko had five points in five games played against the Predators this season.