It was the second meeting between the sides in four days following City's 2-0 win at Gresty Road on Wednesday in the Johnstone's Paint Trophy.
The hosts clearly took confidence from that result, despite exiting the competition 3-2 on aggregate, as they started the brighter and took the lead in the 12th minute.
Leon Clarke got the wrong side of Adam Dugdale to meet Cyrus Christie's diagonal ball and chest down before firing home from 12 yards.
But the visitors drew level just four minutes later when Mark Ellis volleyed Gregor Robertson's corner into the net.
The Sky Blues looked to be having the better of the second half and Robertson was required to head Carl Baker's stunning strike off the line.
But Clayton's 76th-minute introduction proved the turning point as he lobbed Joe Murphy with his first touch after being teed up by Byron Moore, just two minutes after coming on.
- Sports & Recreation
- Gregor Robertson