The draw leaves Chelsea in real danger of missing out on Champions League football for next season, with the champions 10 points behind Tottenham in fourth.
There was also a minute's applause prior to kick-off, with the Chelsea players wearing black armbands in memory of Wilkins.
The same pair combined moments later, but Morata was unable to direct his back-flick from the Belgian's cross into Hart's far corner from close range.
The half ended with the home side leading by a single goal. Firstly, he whipped a 25-yard effort past Hart's post and was then denied by the England goalkeeper's outstretched leg after latching onto Hazard's superb pass.
Chelsea's swagger initially continued after the interval, with Willian dragging a shot just wide and Morata heading over from a corner.
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However, West Ham fought back when substitute Hernandez fired home an emphatic finish in the 72nd minute.
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 Blues got the game off to a flying start following their 3-1 disappointing defeat to Tottenham Hotspur last weekend, but all their efforts to ensure they get early goals were unsuccessful. David Moyes's side, though, can see the relegation zone as more of a dim, ugly mirage with every point they accrue in this tightest of battles in the bottom half of the Premier League.
There was an emotional tribute before the game, and after eight minutes of play, to Ray Wilkins, the former Chelsea captain and assistant manager, who died in midweek.
"Sometimes I can not score against another team and sometimes I keep scoring against Chelsea for example".
Morata did not bring his shooting boots this afternoon, and it appears like he has neglected to wear them for much of the second half of the season; Manchester United centre-back Chris Smalling, with two goals, has scored more times than the Spaniard in the Premier League in 2018.