The Blues were pegged back by Javier Hernandez after Cesar Azpilicueta handed them a first-half lead, in what was the first league game Chelsea had failed to win under Conte when leading at half-time.
Chelsea, meanwhile, slip to 10 points adrift of fourth-placed Tottenham in the race for Champions League qualification.
"Sorry to Chelsea fans", Hernandez said with a laugh.
But goalkeeper Thibaut Courtois and forward Pedro, who were both injured the 3-1 defeat to Tottenham last time out, could make a return at Stamford Bridge. West Ham have lost four out of their last six games and are dangerously close to the relegation spot.
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.
In the 51st minute, Moses was presented with a big chance to put Chelsea two goals up from Morata's pass but blasted over the bar.
His goal had goalkeeper Joe Hart, himself a star on Sunday, praising the diminutive striker for his desire following an worldwide call-up on the other side of the ocean.
Gary Cahill is the other player who is back in the starting lineup, donning the armband as he replaces Andreas Christensen.
Described by many as the ideal gentleman, it was at Chelsea where he launched his senior career, scoring 30 goals from only 179 appearances in the Blues midfield.
While Hart has recently returned to the starting lineup and played 1 more league game than Adrián (15 vs 16) this season, neither keeper has truly impressed in East London. They have to win all their games as a top club so don't think for a minute that they will switch off. "They created a lot but we stood strong".