The Bavarians won their ninth straight match in the competition, equalling the record set by Barcelona in 2002-2003.
In a first half that ended goalless, Bayern had a tough time breaking down the Czech champions who looked dangerous on the break for much of the night.
Mandzukic, who replaced a largely ineffective Thomas Mueller in the second half, broke the deadlock in the 65th minute when he slipped past his marker to head a lofting Philipp Lahm cross past Plzen keeper Matus Kozacik.
The win, coupled with Manchester City's victory against CSKA Moscow in the other Group D qualifying match, scuttled any hopes Plzen had of going into the next round.
Plzen set out to attack from the start and never let the Bavarians get into their rhythm during a first half in which Bayern were unable to break down the home side's defence.
After the restart, Bayern picked up the pace and Franck Ribery almost scored in the 57th minute when his shot from just inside the penalty area was deflected by Kozacik.
The German side came close to doubling their lead late in the second half when Mario Goetze only had the keeper to beat but he could not get a clear shot.
Bayern's victory puts them on 12 points in Group D, while Plzen remain without any points.
(Reporting by Michael Kahn, Editing by Clare Fallon)
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