The Germany midfielder grabbed a 74th-minute winner in the one-legged tie against the run of play with a spectacular 30-metre shot after Bremen had missed several chances and rattled the bar through Marko Arnautovic.
Bremen initially allowed Bayern too much space after taking an early lead through Claudio Pizarro and the hosts equalised on 28 minutes when Schweinsteiger struck from close range.
The visitors upped the pace in the second half and Arnautovic missed two golden opportunities and hit the bar with a 30-metre free-kick before Schweinsteiger settled the tie with a thundering drive.
Schalke 04 struggled against FSV Frankfurt but advanced with a 1-0 win thanks to Jose Manuel Jurado's long-range effort while fellow Bundesliga club Kaiserslautern saw Srdjan Lakic hit all three in their 3-0 win over second division Arminia Bielefeld.
Earlier on Tuesday, VfL Wolfsburg eased into the third round with a 3-1 win at fifth division Victoria Hamburg while Koblenz ousted Hertha Berlin 2-1 helped by a stunning goal from 50 metres (video below).
Third division Koblenz's Michael Stahl will long remember the day after giving them the lead over Hertha Berlin with a shot from inside his own half which sailed over goalkeeper Marco Sejna.
Koblenz won 2-1, inflicting the first defeat of the season on second division leaders Hertha, previously the only unbeaten team in the German professional leagues.
Cologne enjoyed respite from their troubled Bundesliga season when they beat second division 1860 Munich 3-0.
Eintracht Frankfurt host Hamburg SV and Borussia Moenchengladbach take on Bayer Leverkusen in the all-Bundesliga clashes on Wednesday.
