However, while Chelsea may see themselves out of the race for a top-four spot, Antonio Conte's team now sit just three points clear of sixth-placed Arsenal.
Chelsea could have taken the lead in West London as early as the fifth minute when Eden Hazard was given room to advance towards the edge of the penalty area, but the Belgian's low strike hit the outside of the post and went wide.
West Ham rallied to put their hosts under pressure but no chances came their way, and it soon became a case of when Chelsea would be able to open the scoring.
However, from the resulting corner, West Ham failed to deal with two high balls and Azpilicueta was perfectly placed to prod into the bottom corner from close range.
At the start of the second half, Willian was given another opportunity but despite showing neat footwork, his low effort towards the bottom corner went wide of the far post.
West Ham responded through Marko Arnautovic - who forced Thibaut Courtois into a smart stop from 12 yards out - but Chelsea remained in the ascendancy with Morata powering a free header over the crossbar from close range.
Victor Moses also went close as the Nigerian drilled a shot marginally off target with his weaker left foot and it resulted in Hammers boss David Moyes throwing on Hernandez, whose future at the club remains uncertain.
However, the Mexican needed less than three minutes to make an impact. Gary Cahill failed to deal with the ball inside the penalty area and after Arnautovic crossed for Hernandez, the substitute swept the ball into the bottom corner from by the penalty spot.
West Ham opted to remain on the offensive but it was Chelsea who continued to create the better opportunities, with Marcos Alonso forcing Hart to tip the ball over the crossbar from long range.
Just before the 90-minute mark, Olivier Giroud rose above the West Ham defence to direct a header towards the far bottom corner but Hart - who had been excellent all match - got across to tip the ball onto the post.
Both teams looked for a winner in five minutes of added-on time but West Ham will be delighted to have secured a point at the home of one of their London rivals and now appear comfortable in the battle to avoid the drop.
CHELSEA (3-4-3): Courtois; Azpilicueta, Cahill, Rudiger; Moses (Pedro 78'), Kante, Fabregas, Alonso; Willian, Morata (Giroud 78'), Hazard
WEST HAM UNITED (3-5-1-1): Hart; Rice, Ogbonna, Cresswell (Evra 64'); Zabaleta, Kouyate, Noble, Fernandes (Hernandez 70'), Masuaku; Joao Mario (Cullen 85'); Arnautovic