The managers clashed after Mancini raced to retrieve the ball and clattered into the Toffees chief, prompting tempers to fray and referee Peter Walton to send both to the stands. It capped a disastrous night for Mancini, whose side slumped to a first home defeat of the season.
Tim Cahill gave the Toffees a 33rd minute lead with a trademark header from Leighton Baines’ cross, before Mikel Arteta rounded things off five minutes from time after a storming run from Jack Rodwell.
Moyes will have been buoyed by the result at Villa Park where Aston Villa were held for the second time in four days, drawing 1-1 with Sunderland.
Fraizer Campbell gave the Black Cats a shock 22nd minute lead, but although John Carew tapped home an equaliser just eight minutes later, Villa were unable to force a winner.
In the other game, David Dunn scored twice against his former club as Blackburn secured a 2-1 victory over Birmingham. James McFadden equalised Dunn’s fifth minute opener before the ex-Blues star headed home El-Hadji Diouf’s corner.