Barcelona and Inter Milan also got wins to keep their leads as Europe's national leagues resumed after an extended international break.
Bayern coach Vincent Kompany said his team's visit to Augsburg was a "priority game" after their only league loss was against Augsburg in Munich last season, but he can't have imagined making a worse start to what ultimately ended in a 2-2 draw — the second time in five games Bayern have dropped points this season.
It began with Bayern playing the ball back from the kickoff to Neuer, who botched his attempt at a chipped clearance and allowed Robin Fellhauer to score after just eight seconds.
Bayern had to wait till the 55th minute for Ismael Saibari to pull level, only for Rodrigo Ribeiro to restore the home team's lead nine minutes later. Nathaniel Brown finally equalised in the 81st.
Bayer Leverkusen, the only team to break Bayern's dominance in any of the past 14 seasons, held on for a 4-3 win in Mainz, continuing their upturn in form under Spanish coach Carles Martínez.
Christopher Nkunku netted a hat-trick to fuel Leipzig's 6-0 demolition of Eintracht Frankfurt.
Miro Muheim scored in stoppage time for Hamburger SV to win at Hoffenheim 3-2, Union Berlin beat promoted Elversberg 1-0 for their first victory of the season, and Stuttgart won at Paderborn 2-0.
In Spain, Rayo fans forced a suspension of their team's match against Athletic Bilbao by repeatedly throwing paper airplanes and other objects onto the field in protest against club owner Raúl Martín Presa.
It was Rayo's first league game of the season at home since city authorities initially closed the stadium over safety concerns. Presa has long been unpopular among Rayo fans since taking over the club in 2011.
Elsewhere, it was business as usual for Barcelona even without injured top scorer Raphinha. Barcelona equalled their best La Liga start with an eighth win in as many rounds.
Anthony Gordon got his first goal since joining Barcelona, Gabriel Jesus scored in his first start and the defending champions eased to a 3-0 win over Getafe.
Atletico Madrid won at Deportivo La Coruna 2-1 to remain in second at five points back.
Vinícius earned a direct red card for pulling the hair of Villarreal's Nathan Saliba, leaving Real Madrid needing to hang on for the final 20 minutes to earn a 1-0 win.
Vinícius was celebrating after scoring when referee Isidro Díaz de Mera was called by the video assistant to review the moments before his goal. The referee waived off the goal and sent Vinícius off for tugging on Saliba's hair.
Yan Diomande scored his first goal since joining Madrid, finding the net early in the second half.
In France, Khvicha Kvaratskhelia came off the bench to lead Paris Saint-Germain's 3-1 win over Le Mans, while Ousmane Dembélé and Nuno Mendes rested for Tuesday's visit to City in the Champions League.
