Karim Benzema's brace helped Real Madrid secure three points at home against Levante.
Eden Hazard was expected to make his debut in Vigo in the opening round in mid-August after joining Madrid for an initial 100 million euros from Chelsea in June.
Harry Kane scored the only goal of the game with a deft finish after pouncing on a disastrous pass by Marcelo.
Corentin Tolisso, Robert Lewandowski and Serge Gnabry had given Bundesliga champions Bayern a comfortable 3-0 lead.
Barcelona are chasing a ninth La Liga title in 12 years as Real Madrid look to make amends for last season.
Real Madrid struggled without Cristiano Ronaldo last season and Eden Hazard will be tasked with helping to fill the void.
Eden Hazard is the gem Zinedine Zidane can build his new team around and brings the kind of stardust Madrid fans
Eden Hazard is widely expected to leave Chelsea for Real Madrid after seven years in England, and the Europa League win the perfect way for him to bow out.
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.
