However, their victory was marred by fan trouble in and out of the stadium.
Reports suggested fans clashed with coins, bottles and even chairs being thrown in the London Stadium.
And a West Ham source has told Starsport the true extent of the fan violence.
“There was a lot of talk in the build up about police and security with the rivalry between the fans,” a source exclusively told Starsport.
“Heading to the game it was difficult to tell who was a West Ham or a Chelsea fan, which was a bit worrying.
“If police aren’t brought into the stadium sooner, it’s likely to happen again when other big clubs visit”
A West Ham source on the Chelsea clash
“For the majority of the game the fans were focused on the match but West Ham’s 2-0 lead sparked trouble near the end.
“With about 10 minutes to go it turned ugly and stewards started to lose control.
“The segregation became less clear, especially away from the seated section.
“One guy tried to climb over the cover to get to West Ham supporters.
“In the area between the bar and the seats it got to the point where there was no gap between the fans.
“Bottles, coins and other things were being thrown back and forth.
“It was a relief for more than one reason when the game was over.”
West Ham’s new home ground has been plagued with trouble since opening in August.
Fans have clashed inside and outside the stadium as the east end club settle in to their new surroundings.
The visit of rivals Chelsea was seen as a potential trouble hotspot by stewards and police.
And the Hammers source has revealed the trouble continued into the streets after the final whistle.
“Leaving the stadium West Ham fans poured back towards Stratford station but Chelsea fans were in among them,” the source added.
“Some lads were literally looking for fights but most ignored the threats.
“Police vans were everywhere and it was lucky they were there when it kicked off near the station.
“A group of about 15/20 fans clashed as police controlled the situation using their batons.
“Three or four fans were pinned up against the wall by police while the bulk of people attempted to avoid the conflict.
“Hopefully it’s a one-off being the first London derby and a cup tie but if police aren’t brought into the stadium sooner, it’s likely to happen again when other big clubs visit.”
The FA are expected to take action against both sides in the near future.