Gundogan sent Xavi's side ahead in the early stages of the La Liga contest, but Jude Bellingham scored twice in the second half to win it for Los Blancos.
The defeat was Barcelona's first of the season, but they have now dropped points in four of their 11 La Liga matches and sit four points behind division leaders Real Madrid in the table.
Gundogan has said that "there was disappointment" in the dressing room after the loss to Carlo Ancelotti's side, but the Germany international claimed that "there should be stronger feelings".
"I don't want to say anything wrong, but I was in the dressing room and there was disappointment, but to be honest, I would've liked to see more frustration and anger," Gundogan told La Liga TV.
"There should be stronger feelings. I did not come here to lose these types of matches. There is also a responsibility from myself to not allow the squad to let things happen. We need resistance."
When asked about the strength of Barcelona's performance, Gundogan said that the display is irrelevant if they are unable to secure a positive result.
"It's worthless to play well if you don't win," he added.
According to Mundo Deportivo, Gundogan's comments were appreciated by Barcelona's coaching staff, with the club agreeing with the sentiment behind his reaction.
The 33-year-old left Manchester City over the summer following the expiration of his contract with the treble winners, ultimately penning a two-year deal at Camp Nou with the option of a third.
Gundogan has been a vital player for Barcelona during the 2023-24 campaign, making 14 appearances in all competitions, scoring one goal and registering four assists in the process.
Barcelona boss Xavi admitted that his side "were not effective" in El Clasico, having failed to make the most of their strong performance in the first half of action.
The Catalan outfit will have the chance to bounce back from the disappointing result when they head to Real Sociedad in La Liga next weekend.
The La Liga champions will then take on Shakhtar Donetsk and Alaves before there is another break in the campaign for the November internationals.