Bellingham, 17, who joined Dortmund from Birmingham this summer, set up the opener for teenage team-mate Giovanni Reyna in the first half.
England international Jadon Sancho also picked up an assist for Erling Braut Haaland's second goal after the break.
Haaland had earlier scored a penalty.
The spot-kick, won by 17-year-old American Reyna, went to a Video Assistant Review, while Gladbach had an appeal of their own dismissed later in the second half for a challenge by Mats Hummels on Marcus Thuram.
Reyna was one of several young players named in the starting XI, including Haaland, 20, Sancho, 20 and Bellingham.