Roberto Mancini's men were well below par at Eastlands, struggling to create chances against a dogged and determined Reading defence.
But Richards finally broke the deadlock after 74 minutes, rising high to head past a helpless Alex McCarthy from David Silva's corner.
That goal proved decisive with Reading, the last Football League outfit left in the competition, unable to threaten Joe Hart's goal in the final minutes as City shut up shop.
More to follow.
Magic Micah: Manchester City hero Richards (second left) celebrates his goal
One of those days: Manchester City star David Silva rues yet another missed chance
Heart in mouth moment: Joleon Lescott (right) deflects the ball just wide of his own goal
Fall guy: Patrick Vieira (left) goes theatrically to ground under challenge from Mikele Leigertwood
No way past: Yaya Toure (right) sees his shot saved by Reading goalkeeper Alex McCarthy
?Stoke 2 West Ham 1: Higginbotham strike books Bolton semi at WembleyFA CUP SEMI-FINAL DRAW: United await potential City clash at WembleyAdebayor pleads with FA to take leniency over City team-mate Toure Clubs will struggle to find the next Giggs and Scholes, says Vieira
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Source: Daily Mail
Source: Daily Mail