Azpilicueta pointed to the black armband which he was wearing in tribute to Wilkins after finding the net for just the second time in the Premier League this season.
Willian forced Hart into a sharp save on 36 minutes after being played in by Hazard, but the hosts scored from the resulting corner.
Marko Arnautovic had West Ham's only glimpse of goal of the first half and was found wanting at the other end for Azpilicueta's goal. His goal-bound effort was well blocked by Gary Cahill, making his first start since 10 March.
Morata went close in the 29th minute as his clever flick from Hazard cross went agonisingly wide.
Moses and Alvaro Morata both sent the ball into the Matthew Harding Stand from good positions, and Morata had a "goal" rightly chalked off for offside for the second time in the game.
And the Mexican striker had a cheeky message for Blues fans after the game.
Kim Jong Un willing to discuss denuclearizing Korean peninsula during Trump meeting
While Trump is "looking forward to the summit", the schedule could be late May or even early June, CNN said. In a separate report , a United States think-tank says North Korea may have started testing a new reactor.
Chelsea threw players forward in the final ten minutes of the game in search of a victor, but the London derby ended in a stalemate.
The on-loan keeper palmed Marcos Alonso's powerful drive over and then tipped substitute Olivier Giroud's header onto the post during a dramatic finale, in which the Hammers ultimately held firm.
Chelsea will probably not qualify for the Champions League this season but with Manchester City having failed to win on Saturday, the Blues are still the reigning champions - for now at least.
However, they have not lost in 11 home Premier League games against the Hammers (W8 D3).
However, his performance today may keep his hopes alive, as he made three excellent saves to prevent Chelsea from taking all the points on Sunday.
Marcos Alonso nearly spared Chelsea's blushes with a thunderous strike, but Hart finished as West Ham's hero with superb saves to keep out that effort and one from an Olivier Giroud header. That said, their form has been so erratic over the past couple of months that in likelihood, they will welcome Liverpool to Stamford Bridge next month with the top four already beyond them.