Bremen, once a dominant force in the Bundesliga, slumped to an inglorious first-round exit when Saarbruecken struck twice in extra time after a 1-1 draw over 90 minutes.
Toothless Braunschweig, newly-promoted to the Bundesliga, had little to offer in their 2-1 defeat to battling Arminia Bielefeld, who took a 36th-minute lead through Sebastian Hille.
Timo Perthel levelled for the visitors just past the hour but Tim Jerat grabbed the deserved winner in the 72nd minute.
Gladbach's exit was more spectacular with third-division Darmstadt snatching the second-round spot with a 5-4 win on penalties.
Hanover 96 scored twice late in their game to edge past fourth-division Victoria Hamburg 2-0.
On Saturday, Borussia Dortmund left it late against fourth-tier Wilhelmshaven, scoring three times in the final 19 minutes to clinch a 3-0 win and book their spot in the second round.
Holders Bayern Munich are in action on Monday against fourth-tier BSV Rehden.
(Reporting by Karolos Grohmann; Editing by Clare Fallon)
