The hosts could have taken the lead when Andy Little advanced on goal, but his effort went wide from close range.
Leitch-Smith needed two attempts to put Vale ahead, with his initial header from Byron Moore's cross saved by Colin Doyle before bundling home the rebound.
Goalkeeper Jak Alnwick ensured maximum points, denying Danny Philliskirk from close range late on.
AJ Leitch-Smith scored the only goal of the game as Port Vale overcame League One strugglers Blackpool.