The Austrian actor won his first best supporting for his role as a smirking Nazi in Inglourious Basterds.
Lincoln, which stars Daniel Day-Lewis as the 16th president of the US, saw nominations for best drama, best actor in a drama for Day-Lewis, best supporting actor for Tommy Lee Jones, best supporting actress for Sally Field and best director for Spielberg, as well as nods for best script and best score.
Moonlight was the main beneficiary of the Los Angeles Film Critics Association awards, where it took home four, including best picture, best director for Jenkins, best supporting actor for Mahershala Ali, and best cinematography for James Laxton.
Other nominees included the late James Gandolfini, shortlisted posthumously for best supporting male for his work in the romantic comedy Enough Said.
Among its nods were best film, best director for Steven Spielberg, best actor for Daniel Day-Lewis, best supporting actress for Sally Field, best supporting actor for Tommy Lee Jones and best acting ensemble.
The Dark Knight picked up best director for Christopher Nolan, best supporting actor for the late Heath Ledger, best cinematographer for Wally Pfister and best original score for James Newton Howard and Hans Zimmer.
Campion's film, also featuring Harvey Keitel and Sam Neill, won three Oscars at the 1994 Academy Awards, including best actress for Hunter, best supporting actress for the 11-year-old Anna Paquin, and best script for the director herself.
The Sessions is up for best actor, for John Hawkes, and best supporting actress, for Helen Hunt.
It won for best limited series, best actress for Frances McDormand, best actor for Richard Jenkins and best supporting actor for Bill Murray.
