Marcel Sabitzer put the visitors in front in the 28th minute with a ferocious shot from 30 metres that went straight into the top corner with Hertha goalkeeper Rune Jarstein frozen to the spot on a sunny afternoon in the capital.
Leipzig, who lost 2-0 to Liverpool in their Champions League round-of-16 first leg on Tuesday, were initially on the backfoot with Hertha missing several good chances before Sabitzer's goal against the run of play.
The Saxony club are in second place on 47 points, with Bayern top on 49 following their 2-1 loss to Eintracht Frankfurt on Saturday. VfL Wolfsburg and Frankfurt are on 42 in third and fourth place respectively.
Hertha are now without a win in their last eight league games and are stuck in 15th place, level on 18 points with 16th Arminia Bielefeld, who sit in the relegation playoff spot.