The hosts came into the clash buoyed by their Club World Cup triumph last Friday. Manager Hansi Flick made three changes from the side that saw off Tigres in Qatar, with Corentin Tolisso, Bouna Sarr and Serge Gnabry all handed starts.
A wet and snowy night made visibility difficult, but Bayern Munich started the game on the front foot, with Eric Choupo-Moting and Robert Lewandowski both having good chances.
It was, however, the visitors who went ahead against the run of play. Manuel Prietl split the Bayern defence with an exquisite through-pass that was converted with a left-footed finish by Michel Vlap.
The first half saw several stops for snow to be mowed off the field, but Bayern Munich continued to mount pressure on Stefan Ortega's goal.
Arminia doubled their lead in the 39th minute when Amos Pieper rose highest to head home a corner kick.
Bayern Munich responded right at the start of the second half. Leroy Sane hit the post after the goalkeeper parried the ball into his path, and Lewandowski scored with a volley off an assist by David Alaba.
Their reprieve lasted just one minute though, as Arminia Bielefeld reclaimed their two-goal advantage right from the restart.
A brilliant break down the left flank saw Andreas Voglsammer play an excellent square ball into the box, and Christian Gebauer was on hand to finish past Neuer.
The end-to-end fixture continued to see chances fashioned at both ends, and Bayern Munich once again halved the deficit in the 57th minute.
The ever-busy Leroy Sane put in a great cross which Corentin Tolisso headed past Ortega. The Germany international had a hand in his side's equalizer with another great cross that was poorly dealt with by Cedric Brunner.
Amidst a sea of bodies in the area, Alphonso Davies struck the ball into the back of the net with a pristine volley from 17 yards to draw Bayern Munich level.
Bielefeld thought they had gone ahead again just two minutes later when a quick break down the left flank ended in another goal. Fabian Klos was, however, pulled up for a marginal offside after consultations with VAR.
Despite fashioning several presentable chances in the dying stages, neither side managed to score the winning goal to ensure the thrilling game ended with a share of the spoils.
On the overall balance of play, a draw was probably the right result. Here, we shall have a rundown of five talking points from the game.
#5 Brave Arminia get unlikely point to boost survival hopes
The difference in quality and pedigree between the two teams meant that nobody gave Arminia Bielefeld any chance heading into the game.
Nevertheless, Uwe Neuhaus did not accord Bayern Munich too much respect and set his team to play on the front foot from the blast of the whistle.
Time and again, the visitors managed to find space behind Bayern Munich's defence, with their precise counter-attack taking advantage of the hosts' extremely high defensive line.
Their clinicality also came to the fore, with three goals scored from their three shots on target. They fashioned some other notable chances but were let down by poor decision-making in the final third.
Arminia Bielefeld currently find themselves in the midst of a relegation battle, and an unlikely point against Bayern Munich has boosted their chances of avoiding the dropzone.
#4 Poor Bayern Munich defence continues to be a problem for Hansi Flick
Bayern Munich's imperious attack means that an abundance of goals are almost always guaranteed to be scored. It is, however, at the other end where issues continue to plague the team.
Having earlier gone 11 games with 18 goals conceded and no clean sheet kept, Bayern Munich responded with a run of three clean sheets in their last five games.
However, against Bielefeld, familiar problems came to the fore once more as the partnership between Lucas Hernandez and Niklas Sule was sliced open with consummate ease on several occasions.
The manner in which lowly Arminia comfortably created chances against Bayern Munich is a cause for concern.
Moving forward, Hansi Flick has to find a way to plug the leaky defence. Otherwise, it could prove detrimental to the club's chances of success.