Julian Nagelsmann's side dominated the game at the Estadio da Luz on Thursday (AEDT) but had to wait until the 70th minute to open the scoring when Sane curled in a free-kick.
Benfica's Everton increased its lead with an own goal before Robert Lewandowski tapped in and Sane put the gloss on the win with a side-footed finish from inside the box.
Bayern is top of Group E on nine points and has a five-point cushion on second-placed Benfica, while Barcelona is a further point behind in third.
Bayern began with a swagger and almost took an early lead when Thomas Muller's ball over the top picked out Sane but he shot wide.
Odisseas Vlachodimos kept out a close-range Lewandowski header moments later as Benfica's defence were put on the back foot by an intense spell of Bayern pressure.
Benfica's best chance of the first half came when Darwin Nunez brought a fine diving save out of Manuel Neuer, making his 100th appearance in the competition, with a curling effort.
Lewandowski had the ball in the net on the stroke of half-time when he got on the end of Kingsley Coman's cross, only for VAR to deny Bayern with the Polish striker deemed to have bundled it in with his arm.
A crisp half-volley from Benjamin Pavard was deflected onto the post by Vlachodimos just after the interval, while Muller's tap in was chalked off by VAR for offside.
Neuer's reflexes were tested again by a curling Diogo Goncalves effort that the Germany goalkeeper somehow kept out of the top corner.
Lewandowski headed over and Roman Yaremchuk shot wide in an end-to-end contest before Bayern finally seized the lead from a free-kick.
Sane was able to get his left-footed effort up and over the wall, although Vlachodimos was slow to react and almost appeared to dive over the ball.
Bayern had a second goal when Everton turned a Serge Gnabry cross into his own net before Lewandowski finished from a matter of inches out after good play from Sane.
Sane notched his second of the game with six minutes left, stroking the ball beyond Vlachodimos after a fine passing move from the Bundesliga leaders.