Italian Jacopo Sala fired the hosts into a 23rd minute lead, thundering in a volley from a pinpoint Paolo Guerrero cross.
Put under pressure deep in their own half, Bayern had constant problems implementing their game plan and had to wait until the 71st minute to level with former Hamburg player Olic scoring from close range seven minutes after coming on.
Olic, who wanted to move in the transfer window and has demanded more time on the pitch, was by far the most active player for Bayern, who again looked lacklustre until the Croat was introduced.
"It was very difficult for us because Hamburg ran a lot and were constantly pressing," Olic told reporters. "They scored and then waited for quick breaks but we put pressure on them and got our chance. It's better to have a point than not."
Dortmund took command of the Bundesliga on Friday by beating Nuremberg 2-0 to move on to 43 points.
Second-placed Bayern, who have now won only one of their three league games since the winter break, have 41 points from 20 matches and are level with Schalke 04, who drew 1-1 with Mainz 05 and also lost ground in the title race.
"I can't blame the team," said Bayern coach Jupp Heynckes, whose side are chasing a treble of German league and Cup titles and the Champions League.
"We played well but the demands of playing at Bayern are higher and so we cannot really be satisfied," said Heynckes.
Borussia Monchengladbach remained fourth after a 0-0 draw at VfL Wolfsburg ended their three-game winning streak leaving them with 40 points.
Hamburg, who have now lost just one game in the last 11 under coach Thorsten Fink, a long-time Bayern player, stay 11th with 23 points.
VfB Stuttgart rescued a 2-2 draw at 10-man Bayer Leverkusen thanks to substitute Martin Harnik's close-range goal in the 89th minute as both teams finished with 10 men.
Bayer's Michal Kadlec was dismissed just past the hour mark and Stuttgart's Cristian Molinaro was sent off in added time.
Hertha Berlin dropped to 15th spot after a 1-0 home defeat by Hannover 96 whose Norway international Mohamed Abdellaoue scored his 11th goal of the season just past the hour.
It was Hertha's third straight defeat since Michael Skibbe took over and left them just two points above the drop zone.
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