Legwand followed with the winning tally. He skated into the high slot, faked a wrister, and then beat Jaroslav Halak with a backhand.
Martin Erat tallied and Rinne made 39 saves for the Predators, who have won five in a row.
T.J. Oshie netted a goal for the Blues, who got 19 saves from Halak.
The Blues outshot the Predators 29-13 in the first two periods, but Rinne kept the game scoreless heading into the third period.
The Predators got the first tally with 7:28 left in regulation. Mike Fisher fed Shea Weber from the left point. And then from the right circle, Weber quickly flipped a pass to Erat, who tipped it home from the crease.
But the Blues tied it on the power play with 3:52 left. David Backes controlled the disc in the left circle, and slipped a backhanded pass to Oshie, who finished with a backhand from the slot.
The Blues doubled the Predators in shots, 40-20...The Blues went 1-for-6 on the power play, while Nashville wasn't able to tally on its lone power play.