Chelsea missed further chances to take a deserved lead as Pedro missed the target from distance, before Costa picked up an accurate Fabregas through-ball, only to fire over the bar.
Norwich had a penalty shout turned down by referee Craig Pawson after the half-hour mark when Willian brought down Robbie Brady in his own box, while Terry blocked a Sebastien Bassong header off his own line on the stroke of half time.
The match continued in the same vein for much of the second half, with Chelsea probing their opponents while failing to break the deadlock.
Ruddy held a 35-yard free kick from Willian, before the English keeper held a long-range Fabregas effort with relative ease.
Chelsea finally broke the deadlock on 63 minutes through Costa, who ended a run of 603 minutes without a goal by placing a finish in the bottom corner of the net following a neat Fabregas delivery.
The hosts came desperately close to a second goal as Kurt Zouma struck the woodwork from a Willian corner, while Pedro fired wide from 20 yards minutes later.
Nemanja Matic almost added a second for the hosts before the final whistle, as the Serbian drove forwards from midfield, only to be denied by another Ruddy save. body check tags ::