Bayern defended deep and allowed Dortmund to ensure a thrilling closing period as substitute Alexander Isak almost levelled the scores but Boateng deflected his close-range effort in the injury time.
The hosts were able to protect their narrow lead to the end to book their berth for the next round.
Elsewhere, Bremen eliminated Freiburg from the competition as two first-half goals smoothed the way for the 3-2 victory on home soil.
Bayer Leverkusen edged Borussia Monchengladbach 1-0 on the sole goal from Leon Bailey and Eintracht Frankfurt craved out a 2-1 over second division side Heidenheim.
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