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The victory saw Barca leapfrog Atletico Madrid into second place on 16 points, two behind leaders Real who they meet in three weeks' time at the Camp Nou.
Three-time European champions Manchester United played out a dour goalless draw away to AZ Alkmaar.
Messi returned from injury to make his second start of the season against Inter Milan in the Champions League as Barca came from behind to win 2-1 on Wednesday.
Real Madrid are top with 15 points from seven matches and Atletico stayed third with one point less.
Real Madrid climbed to the top of the standings with 14 points from six games, one point and one place ahead of Atletico Madrid, whom they face on Saturday.
The Catalans started without all-time top scorer Lionel Messi as well as teenage sensation Ansu Fati and fell behind in the second minute.
It was Frankfurt's first loss in Europe in 16 games stretching back to 2006, and the win put Arsenal top of the group ahead of Standard Liege.
The result means Atletico Madrid are the only team left in La Liga to have won all of their games this season, with Diego Simeone's side hosting Eibar on Sunday.
Sturridge, who was released by Liverpool after his contract expired last month, can resume playing from July 31, but the FA added that it disagreed with the commission's findings and would appeal the decision to dismiss most of the charges against him.
Reyes was under contract at Spanish second tier side Extremadura. Seven second tier games due to be played on Sunday, including Extremadura's match against Cadiz, were postponed until Tuesday.