Borussia Dortmund suffered a surprise Bundesliga defeat at Augsburg in front of about 6,000 fans.
And Caligiuri scored the second after latching on to Florian Niederlechner's pass and holding off Thomas Meunier before finishing.
Erling Braut Haaland almost pulled one back from Jadon Sancho's pass but his shot was saved by Rafal Gikiewicz.
Augsburg, who have won their opening two games, only had 20% possession and their two goals were their only efforts on target until Alfred Finnbogason forced a double stop from Roman Burki deep into injury time.
English 17-year-old Jude Bellingham made his third appearance of the season but the midfielder was replaced after 60 minutes.
Elsewhere, Bayer Leverkusen and RB Leipzig - who are both likely to be contenders for a Champions League place - drew 1-1.