The only sour note on a straightforward assignment for Chelsea came in the second half, when Ruben Loftus-Cheek limped out of the action with what appeared to be an ankle injury.
Maurizio Sarri raised eyebrows with his decision to leave Eden Hazard out of the starting line-up, but he said he had no choice but to rest the Belgian star.
It was a missed opportunity after defeats for Tottenham, Arsenal and United at the weekend and Chelsea were booed off at full-time.
Chelsea will face Eintracht Frankfurt for a place in the Europa League final after the German side beat Benfica 2-0 on Thursday to go through on away goals.
Liverpool are chasing their first English top-flight title since 1990 and a victory over the Chelsea at Anfield would be a big step toward that dream.
Eden Hazard himself admitted after the game that solo effort against West Ham was "something special".
Chelsea may be forced to cash in on Eden Hazard as their number 10 has just over one year left on his contract and has so far refused to extend his deal.
Victory ensured Chelsea moved into third in the Premier League, two points clear of Tottenham and three ahead of fifth-placed Arsenal in the race for a top-four finish.
Germany were put under immense pressure from a Netherlands side seeking a winner but the visitors snatched victory in the 90th minute through Nico Schulz.
Belgium earned a battling 3-1 win over Russia in Brussels as Eden Hazard struck twice for Roberto Martinez's side.
Both Premier League clubs Arsenal and Chelsea were initially drawn to be at home in the first leg.
Eden Hazard said is happy at Chelsea and agrees with manager Maurizio Sarri over his tactics.
Gonzalo Higuain showed why he has a stellar scoring record over the past decade in Spanish and Italian football with two excellent finishes to open his Premier League account.
Premier League: Tottenham Strike Late Again To Climb Above Manchester City, Chelsea Back In Top Four
Tottenham Hotspur beat Newcastle United 1-0 while London rivals Chelsea outclassed Huddersfield 5-0.
Chelsea prevailed 4-2 in the shootout at Stamford Bridge to set up a final meeting with holders Manchester City on February 24th.
Chelsea remain in fourth place, but they are now just three points ahead of fifth placed Arsenal in the race to qualify for next season's Champions League.
N'Golo Kante scored the lone goal of the match for Chelsea in the 51st minute.
Eden Hazard scored his 100th and 101st Chelsea goals as Maurizio Sarri's men moved back into the Premier League top four by grinding out a 2-1 win at Watford on Boxing Day.
