Five men were nominated for the October Barclays Manager of the Month award.
Those votes were combined with those of a panel of football experts to decide the winner.
Klopp’s Liverpool are unbeaten and six points clear at the top after a stoppage-time winner against Leicester City and coming from behind to get four points from matches with Manchester United and Tottenham Hotspur.
Rodgers’ Foxes recovered from a late defeat at Anfield to beat Burnley and then register the joint-biggest win in Premier League history with a 9-0 victory over Southampton.
Smith oversaw successive wins for Aston Villa with a 5-1 triumph at Norwich City and a late home success against Brighton & Hove Albion, before falling to three second-half goals in a defeat at Manchester City.