Bayern beat Borussia Dortmund 2-0 with both goals scored in extra time but the losing side were aggrieved that a Mats Hummels header midway through the second half was not allowed to stand.
Television replays clearly showed the entire ball had crossed the goalline before it was cleared.
Clubs from Germany's top two divisions voted against the introduction of goalline technology two months ago but Germany's football league (DFL) said this week they would review the matter if a club filed another request.
"We submitted this application for the introduction of goalline technology only for the Bundesliga at the earliest possible moment so as to protect referees," Bayern CEO Karl-Heinz Rummenigge said in a statement.
"It is unbearable to see referees who do not have replays, slow motion or even mathematical calculations to be pilloried in public."
Earlier this season Bayer Leverkusen's Stefan Kiessling scored a 'phantom goal' against Hoffenheim when the ball went in through the side netting.
In the March 24 vote by the 36 clubs from the first and second divisions, only half of the top-flight clubs - including Bayern and Dortmund - were in favour of goalline technology while only three of 18 second tier teams backed the idea.
A two-thirds majority is needed and many clubs are against it because of the technology's cost.
This is why Bayern's request refers only to the top division, Rummenigge said.
"From media coverage of the issue it seems cost is the main reason why second division clubs overwhelmingly rejected it," he said.
The English top-flight became the first domestic league to use technology in August and the system worked well throughout the entire season.
Dubbed Goal Decision System (GDS) and developed by the Hawk-Eye company, the system gives referees a ruling within a second, their watch buzzing to tell them when the ball has gone in.
Goalline technology will also be in use at the World Cup in Brazil in June and July.
(Reporting by Karolos Grohmann; editing by Justin Palmer)
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