Goalkeeper Yann Sommer was in inspired form for the home side, frustrating the champion with a string of excellent saves.
Nagelsmann, who joined from RB Leipzig during the close season, will have been largely pleased with the overall dominance exerted by Bayern but his side was unable to make the most of its numerous chances.
Die Roten looked shaky at the start and were deservedly behind in the 10th minute thanks to Alassane Plea, though Robert Lewandowski, who had previously been thwarted three times by Sommer, volleyed home the equaliser to equal his longest Bundesliga scoring streak (11 matches).
More presentable chances came and went after the interval, with Sommer loving his personal duel with Lewandowski, but the best opportunity of all fell to Gladbach as Bayern was arguably fortunate to take a point back to Munich.
Bayern had two lucky escapes early on as Patrick Herrmann shot wide from the edge of the box before then surging on to an exquisite Florian Neuhaus ball only to make a mess of his eventual squared pass to Lars Stindl despite having only Manuel Neuer to beat.
Stindl was picked out by Christoph Kramer, however, with the Gladbach skipper nudging his throughball past Bayern debutant Dayot Upamecano and Plea was on hand to slam past Neuer.
Lewandowski somehow failed to level in the 26th minute when shooting straight at Sommer from point-blank range, but the Pole found his range just before the break as he met Joshua Kimmich's corner with a controlled volley.
Bayern had firmly established dominance and that continued into the second half, with a desperate block by Sommer denying Alphonso Davies a near-certain goal and he then stuck out a foot to prevent Lewandowski getting his second of the day soon after.
But Gladbach began to create openings on the break towards the end and Marcus Thuram should have sealed the three points 13 minutes from time when he only got the faintest of touches on Stefan Lainer's pass across the face of goal.
Upamecano was then lucky to avoid conceding two late penalties for clumsy collisions with Thuram, and Kimmich had a chance in stoppage-time to seal the win but missed the target despite being completely unmarked in the box as the spoils were ultimately shared.