Chelsea, who have won the FA Cup four times in the last seven seasons, were made to sweat on their place in the last 16 and should have won by a bigger margin after spurning a host of chances.
Oscar and Andre Schurrle both hit the post.
But shortly after completing a deserved win, Mourinho’s side were handed the task of beating Quadruple-chasing City, who have scored 110 goals this season and won all 11 home league matches.
Asked for his reaction to the draw, Mourinho, who was celebrating his 51st birthday, told ITV Sport: “No problem. If you want to be the best then you have to play the best, so it’s good for us.
“One team is the end product. One team needs to improve and has a long way to go.”
Chelsea, who lost 3-2 at Stoke last month, were far too strong for the visitors and deservedly went ahead after 28 minutes with Oscar’s stunning free kick which flew over the wall and high and wide of diving Stoke keeper Asmir Begovic.
An outstanding creative performance from Eden Hazard in the Chelsea midfield and Oscar’s goal seemed to justify Mourinho’s decision to allow Juan Mata to leave for Manchester United.
Stoke manager Mark Hughes, a former Chelsea striker, said there were “more positives than negatives” from his side’s performance.
“Obviously it was a fantastic strike, but I didn’t feel it was a free kick,” he added.
“We couldn’t quite create enough clear openings. It was difficult for us but I’m pleased with what we did. We stuck at it and didn’t go under. I’d suggest there were more positives than negatives.”
In the day’s early match, 10-man Sheffield United put their League One relegation worries aside and gave a battling performance to draw 1-1 with Premier League strugglers Fulham at Bramall Lane.
United, who played most of the second half with 10 men after skipper Michael Doyle was sent off in the 53rd minute, went ahead after 31 minutes when Chris Porter scored. Fulham equalised through Colombian Hugo Rodallega 15 minutes from the end.
Porter reacted quicker to a cross from Harry Maguire, getting to the ball ahead of Fulham defender Philippe Senderos to touch home past goalkeeper David Stockdale.
The balance of the match tipped Fulham’s way when Doyle was sent off for an off-the-ball kick at Fulham’s Muamer Tankovic and after both teams had strong penalty appeals turned down, Fulham equalised.
Rodallega made room for himself on the edge of the penalty area and fired wide of goalkeeper George Long.
Fulham coach Rene Meulensteen made 11 changes from the side that lost to Arsenal in their last league match, bringing in a number of youngsters and resting his senior players for their relegation battle.
But it was United, four times winners of the trophy and conquerors of Premier League Aston Villa in the last round, who played the better football.
The lowest-ranked side left in the competition fully deserved to secure a replay at Craven Cottage on February4. The winners will play Preston North End or Nottingham Forest who also face a replay.
(Editing by Ed Osmond)