Despite struggling to get the ball, Cardiff went ahead in the 79th minute when Hart failed to clear a corner, allowing Campbell to nod the ball home and he repeated the feat from another corner seven minutes later.
Alvaro Negredo pulled one back for Manchester City in injury time, but despite a couple of half-chances were unable to snatch a point at the death.
At White Hart Lane, a second Roberto Soldado penalty in as many weeks was enough to give Spurs a 1-0 win over Swansea City.
Spurs grabbed all three points after Andros Townsend went down under the challenge of Jonjo Shelvey on 58 minutes to give away the spot kick.
The result puts Spurs on maximum points after two matches, despite the absence of Gareth Bale, who was believed to have arrived in Spain on Saturday to finalise his world record transfer fee move to Real Madrid.
(Reporting by Josh Reich, editing by Pritha Sarkar)
