Marc Bartra was able to pull one back late on in the first Klassiker of the Bundesliga campaign, but it was not enough to stop Bayern moving four points clear of second-placed RB Leipzig at the top of the table.
Dortmund were more than happy to sit off their opponents in the opening 45 minutes and cede possession, which initially looked to work well enough as they were given the chance to counter-attack.
The champions did not take long to find their rhythm on enemy territory, though, and they were in front 17 minutes in from the first attempt of the evening.
James Rodriguez was given too much time and space to chest down a back-post cross, before laying it off nicely for Robben to curl past the reach of Roman Burki from 16 yards, making him the club's all-time leading foreign goalscorer.
The contest's pivotal moment arrived shortly after the half-hour mark when, either side of a well-saved Lewandowski shot, Andriy Yarmolenko was denied by Sven Ulreich and Shinji Kagawa missed the target by no more than a few inches.
In typical Bayern fashion, those chances would prove costly as Lewandowski flicked Joshua Kimmich's right-sided cross past Burki via a telling touch from Omer Toprak on its way through - his seventh goal in seven matches against his former side.
Dortmund's evening got even worse before the end of the first half due to Sokratis Papastathopoulos limping off injured, but they could so easily have been back in the match had Yarmolenko got the better of Ulreich in his second clear-cut chance of the contest.
Bayern, now unbeaten in 128 Bundesliga matches that they have led at the midway point, would have sealed the points three minutes into the second half if not for a top Burki save to keep out Lewandowski's downwards header from close range.
From the next attack, a lively Christian Pulisic got to the touchline and cut the ball back into a dangerous zone, where Yarmolenko and Pierre-Emerick Aubameyang comically got in each other's way for the chance to come and go.
Lewandowski uncharacteristically wasted another opportunity when played through by Robben, but he appeared to make amends by getting the smallest of touches to Alaba's cross 67 minutes in to seal the win, though the officials later adjudged that the ball went right the way through with no contact.
From keeping five successive clean sheets to begin the campaign to shipping 12 in their last four, BVB left themselves with too much of a mountain to climb after makeshift full-back Batra curled home an impressive consolation in the dying stages.