The 2. Bundesliga high-fliers prevailed 6-5 on penalties, following a 2-2 draw after extra time in the second round fixture at Holstein Stadium.
Both sides converted all five spot-kicks before Munich's Marc Roca had his effort saved and Fin Bartels slotted home the decisive penalty.
Munich strike early
The Bavarians went ahead through Serge Gnabry in the 14th minute after Ioannis Gelios could only turn Thomas Muller's header into the path of the playmaker.
Muller then should have doubled the lead himself when he fired over the bar from point blank range, and they were made to pay for that miss eight minutes before half time.
The champions' defence has been leaking throughout the season with their most goals conceded after 15 games in the league since 1981, and Kiel struck on the break when Bartels beat Niklas Sule in a sprint to fire past Manuel Neuer.
Despite making five changes to the team that squandered a two-goal lead to lose at Borussia Monchengladbach in the Bundesliga on Friday, coach Hansi Flick's Bayern continued to be vulnerable at the back.
Munich went ahead again through a great Leroy Sane free-kick after 48 minutes, but offered up several good chances for the hosts, who found ways past their opponents' shaky backline.
The clock showed 94 minutes when Johannes van den Bergh pumped the ball into the box from the left. Hauke Wahl was left unmarked at the near post and his looping header beat Neuer to find the far top corner.
Kiel, who next face Darmstadt 98, kept their cool in the shoot out to knock Munich out at this stage of the DFB Pokal for the first time since 2000.
'Kiel punished us'
"We lost to the underdogs and they did everything they could," admitted Muller after the game. "We had chances in the first half, but we made mistakes and Kiel punished us.
"They were very efficient but we should have won the game. We are out and it is bitter."