Chelsea midfielders Willian and Eden Hazard were the game's most creative forces and were both involved in the few scoring opportunities before their team finally struck.
Hazard linked up cleverly with Willian, the Brazilian bringing a good save from Joe Hart.
From the subsequent corner, however, West Ham switched off and allowed Azpilicueta two bites before he jabbed the ball in.
Willian squandered a good chance early in the second half, shooting wide of the far post, and Chelsea were made to pay when Marko Arnautovic set up Hernandez to score only a few minutes after the Mexican striker had come on.
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.
"We did well in the first half apart from the goal which was really poor," Hart told Sky Sports.
"They are the champions, let's not forget. They created a lot but we stood strong."
With six games left, the draw left West Ham in 14th place, only six points short of the target of 40 for survival that their manager David Moyes has set them.
Chelsea's prospects of catching fourth-placed Tottenham, who are 10 points ahead of them, now look even more remote.
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.
Arsenal's Danny Welbeck celebrates his match-winning goal against Southampton at the Emirates Stadium on Sunday.
Photo: PA
Southampton's defeat kept them in the bottom three, three points behind Crystal Palace but with a game in hand.
Shane Long gave them the lead after 17 minutes, nipping in ahead of defender Shkodran Mustafi.
Arsenal equalised 11 minutes later through Pierre-Emerick Aubameyang, playing a one-two with Danny Welbeck.
Welbeck then cut inside onto Alex Iwobi's pass to give Arsene Wenger's side a barely-deserved halftime lead.
Substitute Charlie Austin equalised barely a minute after coming on but with nine minutes left Welbeck headed in Alex Iwobi's cross.
In a wild last few minutes Mohamed Elneny and Southampton's Jack Stephens were sent off in separate incidents.
AAP
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