Mandzukic gave Bayern the lead in the 50th minute before Thomas Muller (72) made it two.
Stuttgart played the final 10 minutes a man light after Martin Harnik was shown a second yellow card.
Bayern now have 48 points to lead comfortably from second-placed Bayer Leverkusen, who could only draw against Freiburg yesterday. Stuttgart stay on 25 points, four behind sixth-placed Schalke.
Dennis Aogo scored one and set up another as Hamburg beat nine-man Werder Bremen 3-2 to collect their first victory of 2013.
Assani Lukimya gave Bremen the lead after nine minutes before Heung-Min Son equalised in the 22nd minute.
Aogo pushed Hamburg ahead in the 46th and Artjoms Rudnevs made it 3-1 after 52 minutes.
Sokratis Papastathopoulos pulled a goal back for Bremen in the 54th minute before captain Clemens Fritz was sent off after receiving a second yellow in the 80th minute.
The visitors' numbers were further reduced in injury time as Marko Arnautovic also saw red.
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