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.
Jarvis, who was proving a handful down the right, sent a pass across the face of goal to Cole who controlled the ball, turned and stabbed it into the bottom right corner.
With just goalkeeper David Marshall to beat, Cole would have had a second after being played in by a beautiful pass but his control deserted him.
Diame was becoming increasing influential and Cardiff were relieved when his shot, which deflected off struggling full-back Matt Connolly, crept around the right post.
Cardiff made a bright start to the second half with Fraizer Campbell prominent and a frantic goalmouth scramble ensued with West Ham prevailing through sheer weight of numbers.
The match was evenly balanced now as it swept from end to end, Cole again undermined by his control as he took sight of goal.
Once more Cardiff were skating on thin ice as Marshall almost pushed a pass to Maiga into the path of Nolan.
Downing arrived with 20 minutes left and he soon injected life into an increasingly lethargic West Ham by dashing forward with the ball.
He then teed up Nolan at the far post only for Ben Turner to intervene at the critical moment.
The Hammers finally produced their second with 15 minutes remaining when Mark Noble launched a crafty run along the edge of the area before finding Nolan who expertly swept the ball into the net.
Cardiff substitute Nicky Maynard almost pulled a late goal back for the visitors, but he scooped his shot over the bar from close range after coming under pressure from Collins.