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Atalanta goalkeeper Juan Musso saved a penalty and a follow-up attempt but he couldn't keep out an own-goal by Davide Zappacosta from the resulting corner as Leipzig salvaged a 1-1 draw.
Villarreal took the game to Bayern and should probably have won by a larger margin given their clear opportunities.
A sold-out Santiago Bernabeu stadium watched silently as a much-improved Barca under the command of new coach Xavi Hernandez dominated from the start.
Real Betis defeated 3-2 Zenit St Petersburg after scoring twice in the first 20 minutes.
Bayern Munich's Polish striker Robert Lewandowski won the FIFA Best Men's Player award for 2021 with Barcelona's Spanish midfielder Alexia Putellas winning the FIFA Best Women's Player prize.
Xavi probably could not have asked for much more in his first game managing a struggling Barca team who are sixth in the standings, eight points behind leaders Sevilla.
Ramos, 35, ran down his contract and, according to Spanish newspaper Marca, was unable to agree an extension after Real offered him a one-year deal on a 10 per cent pay cut which he rejected.
Benzema, who has scored 29 goals in all competitions for Real this season, has not played for Les Bleus since 2015 after a blackmailing scandal over which he faces a trial in October.
Barcelona moved up to second place with 75 points.
Atletico lead the table on 73 points but third-placed Barca, on 71 after beating Villarreal 2-1 earlier on Sunday, can take over first place if they beat Granada on Thursday in their game in hand.