The match lived up to its billing and the result left Bayern and Leverkusen level on 10 points at the top of the standings.
Palacios thundered in his spot kick after Alphonso Davies had brought down Jonas Hofmann. Bayern had taken the lead in the 86th minute through Leon Goretzka, who tapped in after fine early work from substitute Mathys Tel.
“I was happy with the start but after that Leverkusen were better. Then in the end of the first half we had some huge chances to score,” Bayern coach Thomas Tuchel told reporters.
“The result is fair. But if you take the lead in the 86th you should get the three points even if a draw would be the fair result.
“We struggled once more to maintain the level throughout, to remain stable. Our attacks became shorter and more hectic. Even at times when we struggle we have to be more compact,” Tuchel said.
An unmarked Harry Kane handed Bayern an early lead when he headed in at the far post for his fourth goal of the campaign.
The visitors bounced back and started pressing high, with Bayern struggling to keep possession in an entertaining first half.
The pressure paid off as the visitors earned a free kick at the edge of the box. Alejandro Grimaldo curled it perfectly over the wall and into the top corner in the 24th.
Striker Victor Boniface then rode roughshod over the Bayern defence for some 15 minutes, missing chance after chance to go in front.
But Leverkusen also had keeper Lukas Hradecky to thank for taking them level into the break after denying Bayern four times towards the end of the first half with a string of superb saves.
With Leverkusen back in control of the game, Florian Wirtz hit the post after combining well with Boniface in the 79th and the Nigerian, who has scored four times in the league, then missed an opportunity of his own seconds later.
But it was Bayern who scored with Tel weaving past the Leverkusen defence to set up Goretzka.
Leverkusen, however, had the final say with Davies’ sloppy defending earning them a last-gasp spot kick after a VAR review. Palacios stepped up to secure a point for his team.