Mark Noble pumped a long ball into the Chelsea area and Blues captain Gary Cahill responded with a mistimed header that dropped for Marko Arnautovic.
Chelsea were booed off at home, which was undeniably ungracious given the neat performance they displayed, as West Ham secured a draw with the score split one each for either side.
The home side had dominated the London derby for 73 minutes, leading for nearly half that time after Cesar Azpilicueta poked home in following a short corner.
Willian had a fierce drive diverted behind before Morata saw a goal ruled out after straying into an offside position to receive Hazard's incisive pass.
There was also a minute's applause prior to kick-off, with the Chelsea players wearing black armbands in memory of Wilkins.
The Hammers grabbed a battling point at Chelsea on Sunday, in which Hart impressed, to give themselves a healthy six-point cushion above the relegation zone.
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West Ham are next in action when they host Stoke City on April 16.
Hart, keen to restore his reputation as England's best goalkeeper before the World Cup finals, made fine saves from Marcos Alonso and substitute Olivier Giroud late in the game.
His introduction to the game was certainly key for West Ham to get their goal, with Arnautovic struggling as a lone striker. Since my first day here, they have supported me well.
West Ham, which had been much improved in beating Southampton 3-0 a week ago, offered little until Hernandez's intervention soon after coming on.
The point earned against Chelsea put West Ham in a better position to avoid relegation, but the last six games of the season will be hard to overcome.
"I was thrilled to get in at half-time just a goal down, but we were always in with a chance of getting something out of the game".
"How many times have I said we didn't take the chances?" But like the last few months, we didn't and we [conceded]. "But I think this game describes very clearly our season".