Two weeks from an epic Champions Cup quarter-final that Bath came from 21 points down to win 43-41, the sides played out another classic at Franklin's Gardens.
Despite Bath leaving out most of their leading players ahead of next week's European semi-final against Bordeaux-Begles, Northampton were pushed all the way and needed a hat-trick from England wing Tommy Freeman.
Smith's try after just two minutes set the scene for a pulsating encounter.
Freeman crossed twice and Ollie Sleighthome scored the first of his two tries as Saints secured a bonus point before half-time to lead 26-14.
But Bath always remained in touch with tries scored by Louie Hennessey, Arthur Green, Tom Carr-Smith, Tom de Glanville and Archie Griffin before Kepu Tuipulotu's score and a Ciaran Donoghue conversion levelled the scores at 38 all five minutes from time.
There was time for one last twist as Enoch Opoku-Gyamfu was penalised for a high tackle on Henry Pollock, following a TMO check, allowing Smith to snatch the latest thriller between these two sides.