Sam Allardyce's men claimed all three points against the newly-promoted side as the former Liverpool midfielder struck first before the skipper wrapped up the win
Former Liverpool midfielder Cole gave the hosts the lead after linking with Matt Jarvis before Nolan struck in the second half to wrap up all three points for Sam Allardyce's side.
Cardiff was without one close-season signing in the form of striker Andreas Cornelius, who missed out with an ankle injury, but fellow new boys Gary Medel and Steven Caulker made their Premier League debuts for the Welsh side.
James Collins started for the visitors despite fitness worries, while recent additions Stewart Downing, Adrian and Razvan Rat were all on the bench.
The home side began brightly and were rewarded after 12 minutes when Cole slotted home well from inside the area.
The midfielder was played in after good work from Matt Jarvis on the left, with the former Liverpool man turning and firing past David Marshall.
Despite the club's rude awakening to Premier League life, Cardiff fought back well and could have been awarded a penalty shortly after when Fraizer Campbell went down under Winston Reid's challenge, but referee Howard Webb waved away home appeals.
With West Ham pouring forward, Mohamed Diame saw his deflected effort edge past Marshall's left post seven minutes from the break as West Ham enjoyed a period of pressure late in the half.
Diame also had the first chance of the second half, shooting over from the edge of the area 26 seconds into the period, with Marshall then tipping Modibo Maiga's strike narrowly past the far post after 61 minutes.
However, after 75 minutes, the home side went two goals ahead thanks to a sweeping effort from Nolan inside the area.
After Mark Noble had jinked forward, he slipped the ball to Nolan inside the penalty area, and the club captain swiveled, before curling the ball past a helpless Marshall.
Nicky Maynard should have pulled a goal back five minutes from time, but the striker could only stab over from close range, ensuring West Ham a comfortable conclusion to a straightforward victory.