McCarthy netted the only goal of a thrilling game, which saw home midfielder Tugay and Citizens skipper Richard Dunne sent off.
Both sides started brightly, with Martin Petrov warming the gloves of goalkeeper Brad Friedel on eight minutes with a long-range strike which forced a smart diving save from the big American.
And just five minutes later, Rovers took the lead. David Bentley's deep corner from the left was headed back into the mix by Chris Samba, Roque Santa Cruz's header was brilliantly kept out by Kasper Schmeichel, but McCarthy was on hand to tap home the rebound.
Rovers ought to have extended their lead five minutes later courtesy of another Bentley set-piece. The recalled England man delivered a delightful free-kick, but after losing his marker, David Dunn sent his free header over the bar.
The home side were in full control and threatened again on 27 minutes. Santa Cruz's downward header from Stephen Warnock's cross was tipped over the bar by Schmeichel.
And from the resulting corner, the Paraguayan striker got the faintest of touches to Morten Gamst Pedersen's delivery, but as the ball bounced back off the far post, Dunne headed against his own bar, before City eventually scrambled it away for another corner.
Ten minutes later, Bentley slipped in Santa Cruz, but after taking it wide of the City keeper, the chance had gone.
Rovers' final chance of the opening period arrived on the stroke of half-time, as neat build-up play between Santa Cruz and Bentley ended with a low Pedersen drive which was comfortably saved.
Rovers continued where they left off after the break. On 50 minutes, McCarthy and Santa Cruz linked beautifully, but after turning Micah Richards inside and out, the former Bayern Munich marksman just couldn't get his shot away.
The game descended into something of a farce shortly afterwards, as referee Mike Dean brandished a string of yellow cards to the home side and with Tugay collecting two of them, the former Turkey international was sent for an early bath.
Ten minutes later and after Brett Emerton broke brilliantly down the right, Santa Cruz was tugged back by Dunne in the centre and the City captain was dismissed.
Pedersen's free-kick then deflected wide off a City defender and from the resultant corner, Samba headed inches over.
Schmeichel had to be alert to deny a trio of shots from Emerton, as well as a one-on-one with Pedersen, as Rovers dominated the closing stages to stretch their unbeaten run to 14 competitive matches.