The draw cut the Bavarians' lead at the top of the Bundesliga to five points. Meanwhile, their Bielefeld-based opponents remain in 16th place, level on points with 15th-placed Hertha Berlin.
Hansi Flick's men began the game on the front foot in the unrelenting snow. However, they could not find any real opening as the weather conditions slowed down their favored high-tempo game.
Arminia Bielefeld capitalised on Bayern Munich's lack of intensity to open the scoring with less than 10 minutes played, courtesy of a fine finish by Michel Vlap.
The visitors grabbed their second goal through a corner to end the first half with a shock 2-0 lead over the reigning champions.
The snow was cleared off the pitch at the interval, and that seemed to work wonders for Bayern Munich.
The Bavarians immediately put the visitors under loads of pressure with waves of attacks before they were finally rewarded for their troubles.
Robert Lewandowski halved the deficit with a sublime volley into the bottom corner just four minutes after the restart.
However, the hosts were caught napping at the back once again, and Arminia restored their two-goal lead less than a minute after having it threatened.
Bayern Munich did not ease the pressure and earned dividends when goals from Tolisso and Alphonso Davies leveled the score with 20 minutes to play. They kept on pushing for a late winner but failed as the visitors did well to hold on to the draw.
Here are the Bayern Munich player ratings from the encounter:
Bayern Munich player ratings
Manuel Neuer - 4/10
The Bayern Munich captain was left exposed by his defence throughout the game. All three of the visitors' shots on target ended up at the back of his net.
There was nothing he could do about any of the goals as they were all from high-calibre shots.
Bouna Sarr - 5/10
Bouna Sarr and Niklas Sule were at fault for conceding the first goal as they completely failed to close down on Vlap in the box.
Sarr was beaten by Andreas Voglsammer far too easily on his flank. He was hauled off just before the hour mark after a rather disappointing show.
Niklas Sule - 5/10
Niklas Sule was too passive in his marking of Vlap in the build-up to the opener. He then failed to deal with the corner which led to Arminia's second goal.
He was solid in the second half though, as he ended the game with the second-most passes and a team-high three tackles.
Lucas Hernandez - 6/10
Lucas Hernandez was the most assured player in the Bayern Munich backline.
The World Cup winner completed all ground duels he attempted and won two tackles as well as two interceptions.
Alphonso Davies - 6.5/10
Alphonso Davies was a thorn in the flesh of the Arminia defence as he repeatedly charged at their players down his flank.
His goal to make the scores 3-3 was a testament to his abilities as the Canadian struck a delightful volley in the 69th minute of the game.
David Alaba - 7/10
Perhaps the most inspired player for Bayern Munich, David Alaba provided the assist for the Bavarians' first goal with a well-timed dink over the top for Robert Lewandowski.
The versatile Austrian created a game-high four chances as he constantly tried to be the creative spark for the defending champions.
Corentin Tolisso - 6.5/10
Corentin Tolisso filled in for the injured Leon Goretzka but looked off the pace for most of the first half.
However, he came to life in the second half and even scored Bayern Munich's second goal of the night. The midfielder timed his run to perfection to meet Leroy Sane's cross and head the ball home.
Leroy Sane - 6.5/10
Leroy Sane endured a mixed spell as he failed to lay a foothold in the game.
The former Manchester City winger managed to provide the assist for Tolisso's goal with a beautiful cross into the box. However, he was unlucky not to score one himself when he clipped the crossbar early in the second half.
Kingsley Coman- 6/10
Kingsley Coman was the spark in the Bayern Munich attack. The Frenchman repeatedly took on opposition defenders and was successful with most of his attempts. He completed a game-high seven take-ons from an attempted 10 and tormented the right side of the Arminia defence.
Eric Maxim Choupo-Moting - 5/10
Eric Maxim Choupo-Moting failed to have a meaningful impact on the game. Although he showed his fancy footwork on a couple of occasions, he was isolated for large spells of the game.
Choupo-Moting completed the least number of passes for Bayern Munich, with just 11 passes in 90 minutes.
Robert Lewandowski - 6.5/10
Robert Lewandowski began Bayern Munich's comeback with a sensational volley at the start of the second half.
However, he was guilty of some wasteful finishing as he had two big chances to win the game but failed to deliver.
Joshua Kimmich - 6/10
Joshua Kimmich was brought on for the struggling Sarr and delivered superbly. Bayern Munich conceded no goals when he was on the pitch, and his seemingly undying energy levels were very useful for the Bavarians while going forward.