Cunningham headed home just before half time while Emmanuel-Thomas sealed the deal 13 minutes from time as Taylor, who admitted before kick-off that he would understand if the sack came, lost again.
Danny Kedwell pulled one back for Gillingham from the spot four minutes after Emmanuel-Thomas’ effort after Aden Flint was judged to have handled but it wasn’t enough.
Gillingham welcomed League One leaders Bristol City to Priestfield having won just once in nine games in all competitions and were dominated early on.
City threatened through Emmanuel-Thomas’ drive and Kieran Agard’s shot on the turn and were nearly gifted a 36th-minute opener when Leon Legge almost scuffed a clearance into his own net.
Just four minutes later Cunningham headed home from close range following a Scott Wagstaff cross and Emmanuel-Thomas doubled the lead in the 77h minute with a precision finish from 25 yards.
Kedwell pulled one back from the spot after Flint’s misfortune but Gillingham could not find an equaliser.