It was the first two-goal game for Chase, who has 1,871 penalty minutes and just 17 goals in 426 NHL matches. He also picked up an assist.
Expansion Nashville is 0-8-1 in its last nine home games.
Blues rookie goaltender Brent Johnson, the grandson of Hall of Famer Sid Abel, made 30 saves and is 3-0-0 in his first three career starts.
Chase opened the scoring with a wrist shot over Mike Dunham's glove from the left circle at 18:26. At 18:59, he made it 2-0 on the rebound of Scott Young's shot. Mike Eastwood, who has eight points in six games against Nashville, assisted on both Chase goals.
Jim Campbell connected early in the second period, putting the rebound of his own shot off the right post past Dunham. Craig Conroy made it 4-0 on a shot off Nashville defenseman Bob Boughner.
Young got his 200th career goal off Chase's feed in the third period.
Vitali Yachmenev spoiled Johnson's shutout bid with 9:34 left, tipping in Jamie Heward's point shot.
While the Predators have been woeful at home, the club is 5-2-1 in its last eight on the road and starts a four-game Western swing Thursday at Los Angeles.
St. Louis snapped the old club record of five consecutive road victories, set from Jan. 15- 30, 1997.
Goaltender Jim Carey, the former Vezina Trophy winner signed by the Blues on Monday, joined the team but was a healthy scratch. Jamie McLennan served as Johnson's backup.
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