Messi on song as Barca get off to best start in La Liga history
Messi's 11th double of the season left him one goal shy of Gerd Muller's record of 85 for club and country in a calendar year, set in 1972.
The 25-year-old, who could become the first player to win the FIFA world player of the year award four times, has scored 28 times this season, including a league-leading 21 goals.
Real, meanwhile, extended their dominance over Atletico Madrid with a 2-0 victory, keeping alive a run of not losing to their city rivals in 24 straight games, dating back to 1999.
Atletico went into the game after their best start to a season with 11 wins in 13 rounds. But any hopes of ending 13 years of misery were crushed when Cristiano Ronaldo scored an exquisite free kick in the 16th minute.
Atletico remain in second place after their second loss of the campaign. But Barcelona are six points clear, as third-place Madrid closed to within five of their derby rivals.
Torino played more than 50 minutes at a numerical disadvantage after Kamil Glik was sent off following a horrific challenge, with the score still 0-0.
Juventus midfielder Andrea Pirlo blazed a penalty over the bar at the end of the first half.
Goetze levelled in the 74th minute, when he controlled Marco Reus' corner with his chest and struck the dropping ball sweetly inside the far post.
Toni Kroos broke the deadlock after a period of concerted Bayern pressure in the 67th minute, eluding Mats Hummels and Neven Subotic on the edge of the area before striking the ball perfectly with his left boot to send it beyond the helpless Roman Weidenfeller in goal. The latter made three outstanding saves to save the point late in the game.
Bayern defender Holger Badstuber was carried off in the first half with damaged cruciate ligaments in his right knee.