Joshua Kimmich in the 64th minute, Raphael Guerreiro in the 69th and teenager Lennart Karl in the 81st decided the top-v-bottom duel which started 15 minutes late due to Bayern's delayed arrival.
Winless Gladbach had Jens Castrop sent off in the 18th minute but held on until past the hour-mark before Bayern's pressure and patience eventually made the difference. A penalty miss by Kevin Stöger at 2-0 did not help.
Milan are the only other team from Europe's top five leagues to win their first 13 matches, in 1992-93. Bayern can get sole possession of the record on Wednesday with a German Cup victory in Cologne.
"The red card made it into a game of patience," Kimmich told Sky TV before talking about the winning streak.
"It is not coincidence, it is the reward for hard work. We have many players who can decide games. Everyone is competing for the team," he said.
RB Leipzig remain five points back in second after they handed Augsburg their biggest Bundesliga home defeat with a 6-0 scoreline. Teenager Yan Diomande opened the scoring and registered two assists.
Eintracht Frankfurt rebounded from a 5-1 Champions League debacle against Liverpool with a 2-0 over St Pauli, who suffered a fifth defeat in a row.
Socceroos Jackson Irvine and Connor Metcalfe both featured form the bench but only after St Pauli were already two-down.
A 96th-minute goal by Max Beier earned third-placed Borussia Dortmund a last-gasp victory at home to Cologne.
Hoffenheim got their first points at home this season beating lowly Heidenheim 3-1. Wolfsburg ended a run of four straight defeats with a 1-0 victory at SV Hamburg.
