Vieira, a four-time FA Cup winner with Arsenal, struck in the 37th and 58th minutes before substitute Tevez, Dzeko and Richards finished things off in the final six minutes.
County had dominated early on, hitting a post through Karl Howley, but the Barclays Premier League side upped their game to run out comfortable winners.
VOLLEY HOME
Ivan Klasnic's second FA Cup goal in four days secured Bolton Wanderers a 1-0 victory over Fulham and a quarter final trip to Birmingham City.
The Croatian striker scored the winner in Wednesday's fourth-round replay against Wigan Athletic and he was on hand again here to volley home from 12 yards.
Fulham welcomed Bobby Zamora back into action after five months out with a broken leg but they could not find a breakthrough in a fractious encounter at Craven Cottage.
LATE RALLY
Leyton Orient handed Arsenal a reality check as the League One side battled to a famous 1-1 draw in their FA Cup fifth-round tie at Brisbane Road.
The Gunners made 10 changes from the side which had beaten Barcelona in midweek, but lacked a cutting edge in attack before they eventually made the breakthrough when Tomas Rosicky headed in on 53 minutes.
However, Orient produced a late rally and earned themselves a replay when substitute Jonathan Tehoue drilled home with a minute left to keep alive the dream of a quarter-final at Manchester United.
SUNDAY'S FA CUP RESULTS
Manchester City 5-0 Notts County
Fulham 0-1 Bolton Wanderers
Leyton Orient 1-1 Arsenal