Germany midfielder Leon Goretzka looked to have secured Bayern the three points when he scored with four minutes remaining, but a Video Assistant Referee review in injury time showed a foul by Alphonso Davies, bringing Palacios to the spot.
The Argentinian converted to ensure his side remained unbeaten this season.
A frustrated Bayern captain Thomas Mueller approached the referees to complain after the match over “two interesting decisions before the two goals we conceded.”
“We’re playing a contact sport,” Mueller told DAZN, calling the late spot-kick “a very soft penalty.”
“In England, the referee wouldn’t blow the whistle.”
Bayern coach Tuchel agreed, saying, “There was contact but we can’t forget that the referee didn’t blow the whistle (on the pitch).
“In the 94th minute, in a top game, it’s too little. It was super soft.”
Xabi Alonso’s Leverkusen remains ahead of reigning champions Bayern at the top of the table on goal difference, with both sides on 10 points from four matches.
“A lot happened and for us a draw is good. We want to keep going,” Alonso said.
Kane scores early
In the week that Bayern unveiled a statue of club legend Gerd Mueller, who scored 365 league goals for the Bavarians, outside their Allianz Arena home, current number nine Kane had his say early.
From a Leroy Sane corner, Kane shed marker Granit Xhaka and headed in at the far post.
Despite scoring 19 goals to go unbeaten in their first four competitive fixtures this season, Leverkusen’s attack failed to click early and it took a set-piece to get on level terms.
Mueller gave away a free-kick on the edge of the ‘D’, allowing Grimaldo to unleash a stunning, dipping strike past the helpless home goalkeeper Sven Ulreich.
Mueller later complained about the decision, claiming, “If I had just fallen over, we wouldn’t have conceded that goal.”
The equaliser ignited Leverkusen, the visitors’ fluent and precise interplay — in particular the budding partnership between Florian Wirtz and Victor Boniface — pushing Bayern onto the back foot.
Collecting a ball from Wirtz, Boniface had the ball in the back of the net, but the goal was scrubbed off for a narrow offside.
Leverkusen picked up in the second half from where they ended the first, frequently looking dangerous as Wirtz found space between the lines.
Wirtz had a go himself with just over 10 minutes to go, his effort beating Ulreich but rebounding off the inside of the post.
Tuchel brought on forward Mathys Tel and the French teenager, who scored a late winner from the bench against Borussia Moenchengladbach before the international break, was again decisive, finding Goretzka directly in front of goal.
Goretzka cooly rolled the ball home, sending his teammates, who thought the match was done, spilling off the bench to celebrate.
With Leverkusen pushing for an equaliser, Davies brought down Jonas Hofmann in the box and Argentinian World Cup winner Palacios converted the penalty to snatch a point.
On Wednesday, Bayern begin their Champions League campaign at home against Manchester United.