Meanwhile, with Sturridge's departure confirmed, Chelsea stepped up its bid to lure Newcastle striker Demba Ba to London.
The Upton Park side, who are bottom of the Premier League, signed forward Demba Ba from Hoffenheim on Friday.
BBC: West Ham confirm loan deal for Tottenham's Robbie Keane
West Ham have also recruited defender Wayne Bridge, midfielder Gary O'Neil and striker Demba Ba during the January transfer window.
BBC: West Ham confirm loan deal for Tottenham's Robbie Keane
New signing Demba Ba and Belgian midfielder Eden Hazard had given Chelsea a comfortable advantage at the break as the Blues dominated.
West Ham do have several attacking options, however, including new signing Demba Ba, Carlton Cole, Frederic Piquionne, Victor Obinna and Freddie Sears.
Demba Ba and Jonathan Spector failed to trouble Paul Robinson with close-range headers, before Freddie Sears saw a 20-yard drive held by the Rovers keeper.
Cole and Demba Ba went close for the visitors while Jermain Defoe was denied by Robert Green, who also tipped Gareth Bale's free-kick on to the bar.
Fernando Torres, preferred to new signing Demba Ba, continued his stuttering form by blasting over from inside the area as Chelsea dominated right up until half time.
Quite what Chelsea's prospective signing Demba Ba thought of the team's performance is unknown, but with the Newcastle striker sat in the stands, the Blues flattered to deceive.
Even before the window had opened Newcastle found themselves at the sharp end of that "cut-throat industry" as Chelsea expressed an interest in signing Magpies striker Demba Ba.
"Newcastle United can confirm that Chelsea Football Club has made an official offer for Demba Ba which has triggered the release clause in the player's contract, " the club's website reported.
Robbie Keane had a quiet home debut for the Hammers and substitute Demba Ba hit the post from close in as Grant's side returned to the bottom of the league.
The game turned on an incident just after the break as Chelsea won a penalty when England goalkeeper Joe Hart brought down Demba Ba as he attempted to go round him.
Rafa Benitez's week finished in more positive fashion as Chelsea beat West Bromwich Albion at Stamford Bridge -- Demba Ba getting the only goal of the game in the 28th minute.
The Hammers failed to clinch the loan signing of Hoffenheim's Senegalese international striker Demba Ba in time for him to feature at St Andrew's but Grant said the move is close to going through.
But in the first installment Sullivan discusses Demba Ba, the French-born Senegalese striker who played for West Ham last season, left when the club was relegated, and is now perhaps the hottest attacking player in the Premiership.
FORBES: West Ham Chairman Reveals Demba Ba's Contract Details
The defeat dropped West Ham back into the relegation zone and Hammers boss Avram Grant believed Vidic should have been sent off for a perceived professional foul on Demba Ba rather than the yellow card he received when the visitors were trailing 2-0.
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