Nagelsmann's team, which included big Premier League transfers Florian Wirtz and Nick Woltemade, were completely toothless in attack and error-prone at the back while lacking any game plan as the Slovaks dismantled them to inflict a first-ever away defeat on Germany in a World Cup qualifier.
"That was just not good enough from every one of us," Germany captain Joshua Kimmich said of their third consecutive defeat.
The Germans also lost to Portugal and France in the Nations League in June. Their next World Cup qualifier is at home to Northern Ireland in Cologne on Sunday in Group A, which also includes Luxembourg.
"We have a chance to do things better in three days," Kimmich said. "We have talked about the World Cup before this match but we first have to qualify. Because with performances such as this it won't work with the qualification."
The four-time world champions last won the World Cup in 2014 and have since failed to make any impact internationally. They were eliminated in the group stage at the last two World Cups.
Nagelsmann, who led Germany to the quarter-finals on home soil at Euro 2024, had said on Wednesday that setting high goals and wanting to win next year's World Cup, which will be co-hosted by the United States, Canada and Mexico, was a sign that the team was in good health.
After Thursday's bitter defeat, however, Germany fans are demanding more urgency and aggression from their team as well as immediate changes, including to their vulnerable backline with centre back Antonio Ruediger beaten for both Slovakia's goals.
It will also be key for Nagelsmann to find a playmaker, ready to take on the responsibility of implementing a clear game plan, with Germany badly missing a connection between defence and attack.
"I do believe that the players now feel what is needed (on Sunday). The entire team must carry that energy. If we are more energetic, then we will have a better game on Sunday," said Nagelsmann.