Only Cole's fine early strike separated the teams until Nolan put the result beyond doubt in the 76th minute, but the Hammers would already have been out of sight had they been more clinical.
It was a tough opening to Cardiff's first appearance in the top flight since 1962, but there was enough fight on display to suggest they have a chance of avoiding an immediate return to the Sky Bet Championship.
The Premier League newcomers have spent close to £30million over the summer and they paraded record £9.5million signing Gary Medel at Upton Park, but he was largely anonymous, as was his fellow new recruit Steven Caulker.
Stewart Downing, West Ham's £5million recruit from Liverpool, made his debut in the final 20 minutes and his introduction gave West Ham an extra gear in a more finely-balanced second half.
The Hammers were coasting until the interval after creating several good chances that were wasted by Cole, Matt Jarvis and Mohamed Diame.
Two of them were heavily involved early on, Jarvis floating a clever cross to the far post only for Cole to produce a poor header as West Ham failed to make the most of a promising start that saw the trickery of Modibo Maiga cause problems.
Maiga helped set up a chance for Jarvis that was spoilt by the winger's poor first touch and Cardiff staged a counter attack that ended with Kim Bo-kyung driving high and wide over the crossbar.
Kim had Craig Bellamy in support and the South Korean was scalded by his captain for declining to pass.
Play quickly swept to the other end as West Ham plundered their first goal by capitalising on some indecisive defending.