A Bobby De Cordova-Reid stunner was enough for the Cottagers to reach the fourth round.
The Jamaica international capitalised on a mistake and scored his second goal in as many games after last Sunday's winner during Fulham's 2-1 victory over Arsenal in the Premier League.
Richardson believes his side, who sit bottom of the Championship, will grow after the defeat.
The Millers had little to show for their efforts on the night but came close when Jordan Hugill's attempt was ruled out for offside.
Richardson said: "Congratulations to Fulham. We are disappointed when we lose a game, but great credit goes to the players.
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