Hansi Flick gave Jamal Musiala a start, dropped Leroy Sane and recalled Robert Lewandowski, Serge Gnabry and Kingsley Coman from the win over Lokomotiv Moscow in the UEFA Champions League.
Grischa Promel put the all-conquering visitors on the back-foot on four minutes after propping up at the back-post to head home a corner-kick from Christopher Trommel.
Bayern should've been level four minutes from going behind from a through ball to Gnabry, but the forward was unable to get the desired outcome.
Awoniyi could've finished the first half with a brace at the midway stage after chasing down a ball over the top of the defence but after holding off David Alaba, dragged wide.
An injury forced substitution ten minutes to halftime from the host saw Marcus Ingvartsen make way for striker Cedric Teuchert.
Bayern continued threatening 43 minutes in from a smartly taken volley in the area by Gnabry, but Berlin goalkeeper Andreas Luthe was equal to the task with a smart save.
Luthe was called to duty again ten minutes in the second half with an outstanding save to deny Coman profiting from a one-on-one opportunity spearheaded by Lewandowski.
The Bavarians did eventually level it up midway through the second half after Lewandowski lurked in the area to prod a nicely worked cutback from Coman into the back of the net.
Union could've regain the lead inside the closing minutes from a curling strike by Keita Endo following a counter-attack, but Manuel Neuer went down to make a save.
Both sides had chances to secure the winner as fulltime loomed, but it finished all square with Union continuing their fine start to the season.