Dortmund's earlier 3-1 win at Wolfsburg saw it climb to the summit briefly, but Bayern responded in Sunday's (AEDT) late kick-off to move a point clear ahead of next week's Klassiker.
After failing to forge a breakthrough in the first half of a league game for just the second time this term, Sane finally found the net for Bayern with 71 minutes played.
That was Bayern's 102nd league goal this calendar year – a Bundesliga record – and proved enough to help the hosts bounce back from last week's 2-1 loss at Augsburg.
Reigning champion Bayern had 11 attempts in the first half, six of those on target, without finding a way past inspired Arminia goalkeeper Stefan Ortega.
The visiting keeper did well to keep out a couple of Alphonso Davies efforts and a volley from Thomas Muller, who earlier had a header cleared off the line by Andres Andrade.
Sane continued Bayern's wastefulness by missing two big chances early in the second half, but the Germany international eventually provided the winner.
Collecting a pass from Muller outside the box, Sane whipped a shot past Ortega to give Bayern the goal it craved.
Arminia held Bayern 3-3 in its previous meeting in February, but the league struggler could not find a response this time and was fortunate not to concede a second when Serge Gnabry's shot came back out off the crossbar late on.