The right-back's 74th minute header effort ended the club's 30-year wait to reach the latter stages of the competition. For the Royals, a dogged spirit was on show but they couldn't provide the support to 20-goal lone striker Shane Long to force an upset.
For the hosts, manager Roberto Mancini made four changes to the side that lost 2-0 to Dynamo Kiev in the Europa League on Thursday. Top scorer Carlos Tevez was welcomed back along with Nigel de Jong, Patrick Vieira and Shaun Wright-Phillips as Edin Dzeko, Gareth Barry, Mario Balotelli and Pablo Zabaleta made way.
Royals boss Brian McDermott made three alterations from the team that beat Ipswich Town 3-1 in The Championship on Tuesday. Ex-City starlet Matthew Mills' groin injury ensured he dropped out along with Hal Robson-Kanu and Brian Howard; Jay Tabb, Jimmy Kebe and Jobi McAnuff - who had recovered from an ankle knock - replaced them.
Despite the buzz that surged through Eastlands at the prospect of an all-Manchester semi-final, Reading edged the opening minutes. Wingers McAnuff and Kebe caused repeat problems, with the former forcing a corner five minutes in which came to nothing.
Reading - who were 25 places below their hosts in the football ladder - continued to press from the wings in the 15th minute moments after a delightful David Silva through ball found the marginally offside Wright-Phillips. A swift counter from the free-kick saw a deep Kebe centre find McAnuff, who cut inside Richards and teed up Noel Hunt on the back post, but the ball was just behind the Irishman and he could only head over on the stretch.
Silva's creative presence then began to tell and he came close to opening the scoring on 18 minutes. The slight Spaniard won the ball from Reading enforcer Mikele Leigertwood in the semi-circle and immediately played in Tevez.
The Argentine repeated the favour for Wright-Phillips and the wide man's powerful shot from inside the penalty box was palmed away by Alex McCarthy into Silva's path. Silva then stabbed at the rebound from five yards but could only find the prone goalkeeper who deflected the ball for a corner.
Reading's drive ensured they matched their vaunted opponents in most areas but they struggled to keep a lid on Silva's artistry. A wonderful, 20-yard lofted through-ball on the twist from the £24 million summer signing beat the offside trap and put the on-rushing Toure one-on-one with McCarthy on 28 minutes but the Irishman stood tall and ricocheted the effort wide off his legs.
The match had long since turned into the 'Silva show' and he again went close to breaking Reading's resolve in the 36th minute. The playmaker trapped a long-ball on the right wing - and reminiscent of Lionel Messi's glorious opener against Barcelona in mid-week - he then instantly flicked the ball over befuddled left-back Ian Harte's head only for his low shot to be denied again by McCarthy.
City's class was beginning to tell and they continued to turn the screw as the half ebbed towards conclusion. In thd 41st minute, Toure's long legs and gargantuan frame bundled past Zurab Khizanishvilli on the right wing but the combination of Tevez and Wright-Phillips was crowded out by the scrambling defence from the Ivorian's cross.
City were now quickest to every ball and in the 43rd minute a rare selfish moment from Tevez defused another attack. The striker should have played in Wright-Phillips as the recalled winger charged into open ground in the centre of the penalty area, but he instead chose to shoot into Harte's legs.
Silva's final effort to prize Reading open saw his lobbed pass drop a few feet from Vieira before referee Lee Probert called for the interval.
Reading had been on the ropes at the end of the first-half and were almost floored by an own-goal from centre forward Shane Long. The Irishman skimmed a 46th minute Aleksandar Kolarov corner towards the net, only for Jay Tabb to stick out a vital leg.
The shock seemed to jolt the Royals into life and they soon threatened the City net. In the 50th minute, Noel Hunt sliced a shot wide of Joe Hart's near post from inside the box when found by Kebe.
The tide turned again towards City a minute later but the opener was still elusive. Sharp footwork from Silva saw him dribble past Khizanishvilli and Jem Karacan to find Tevez's whose improvised back-heel from three-yards was pounced on by McCarthy.
Reading them almost constructed their own demise as the previously stoic Khizanishvilli's rushed clearance was charged down by Wright-Phillips in the 55th minute. His cross was cut-back to Tevez, but in a recurring them to the afternoon, his effort was harmlessly scuffed into the turf.
City now had a grip on the match and in the 74th minute they finally went in front through the head of Richards. The defender powerfully rose to meet an outswinging corner from Silva to tee up a mouthwatering meeting with United.
Richards came to national prominence after an expletive-laden interview in 2006 following a goalscoring performance in the F.A. Cup against Aston Villa and he couldn't be blamed for a similar outburst again due to the importance of his strike.
Stung by the concession, Reading began to threaten City in the air. In the 82nd minute, stand-in centre-back Bryn Gunnarsson headed a deep Ian Harte free-kick wide of Hart's far post.
Three minutes later, and substitute Simon Church was inches away from poking a shot in from the goal line following a scramble from a corner.
Nerves were frayed among the home support but City managed to hold on to the slender lead to set up the tie of the season.