Nolann Le Garrec put Racing in front with an early penalty and West failed to make it 3-3 when he missed a straightforward chance from the tee.
Virimi Vakatawa extended the Paris club's lead with a sharp sidestep to squeeze beyond two defenders and dot down for an opening try, which Le Garrec converted after 25 minutes.
La Rochelle applied sustain pressure and although Raymond Rhule had a try ruled out as the ball was not clearly out of a ruck when he scooped it up, West's penalty got them on the board and Gregory Alldritt powered his way over on the stroke of half-time.
West was unable to add the extras and the La Rochelle fly-half was off target with another penalty early in the second half before the ice cool Le Garrec booted Racing into a 13-8 lead.
Ill-disciplined Racing were rocked when Camille Chat and Cedate Gomes Sa were sent to the sin bin in quick succession, the second offence resulting in a penalty try that put last year's runners-up 15-13 up with just under half an hour go.
A disjointed and sloppy La Rochelle side failed to increase their advantage while Racing were two men down, during which time Le Garrec missed from the tee twice.
Pierre Popelin stepped up to miss a penalty at the other end, but O'Gara's men ended Racing's hopes of winning the Champions Cup for the first time, with West diving over to seal it.
