Choupo-Moting double sees Bayern through
Bayern was behind within nine minutes after Mads Pedersen fired the host in front, but a double from Eric Maxim Choupo-Moting and a smart Joshua Kimmich finish put the Bundesliga champions 3-1 ahead.
Dayot Upamecano's unfortunate own goal gave Augsburg hopes of a comeback, but Jamal Musiala and Alphonso Davies both struck late to ensure Bayern's progression into the third round of a competition they have won 20 times, 14 more than any other side.
Despite scoring five against Augsburg, Nagelsmann wanted his side to be even more ruthless, saying: "It's a classic cup game.
"The team created enough chances to make the game clearer earlier on, but unfortunately we missed.
"There were many situations in which we should have scored more goals."
Nagelsmann was thankful for his side's improvement after finding themselves a goal down early on, particularly their dominant period after half-time which saw Kimmich and Choupo-Moting make it 3-1 with 53rd and 59th-minute goals.
"We didn't have a good start, we were very sluggish and didn't accept many duels," Nagelsmann said.
"The 20 minutes after half-time were very, very good. Physically we were on the same level."
When asked why Bayern struggled at the start of the match, Nagelsmann said: "It's not that easy to play against a team that plays every ball straight ahead.
"It's always a balancing act. Do you play football, or do you play the ball in the opposing half?
"We didn't always make the right decision."