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Carlos Edwards re-established their advantage minutes after the break, before Frank Nouble secured the points in the dying stages - albeit under farcical circumstances - and the hosts went on to register their home win since the 2-0 victory over Brighton & Hove Albion 67 days ago.
It was not enough to earn the hosts a half-time lead, as Rovers drew level 15 minutes from the interval.
Goalkeeper Dean Gerken raced out of his area to clear a long pass played from defence, only for the shot-stopper to be beaten to the ball by Rhodes, who passed into an empty net to restore parity.
However, McCarthy's charges re-took the lead just three minutes into the second half, as Stephen Hunt embarked on a mazy run into the opposition box, before the ball fell to Edwards who slotted home from close range.
The Blues made sure of the points on 86 minutes under similar circumstances as they had conceded, with Kean this time culpable for not dealing with a long ball as Nouble capitalised on the shot-stopper's hesitancy to wrap up the points for McCarthy's charges.