The incident took place after Fabio Carvalho's 98th-minute winner
Members of both teams' coaching staff clashed after Fabio Carvalho's 98th-minute winner on 31 August.
Achterberg has been fined £7,000, while Hodges has been fined £6,000.
Both have been banned from the touchline for one fixture and warned about their future conduct.
Achterberg, Liverpool's goalkeeping coach, has accepted he used abusive and/or insulting gestures, while both men have admitted their behaviour was improper.
It follows a review by the FA into an incident where a Newcastle staff member appeared to throw an object towards the Liverpool technical area.
-- Courtesy of BBC Sport