The Germany international opened the scoring in the 38th minute, tucking home after being played in by an eye-of-the-needle pass through the home defence from Emil Forsberg.
The second goal two minutes after the break was all the striker's own work as he burst through a static Monchengladbach back line and finished past Yann Sommer.
Werner's pace was just too hot for the hosts to handle as Leipzig threatened further goals, but they were almost pegged back when Willi Orban deflected Stefan Lainer's cross against his own bar with his chest.
At the other end Sommer pulled off a fine save to deny Marcel Sabitzer.
Gladbach grabbed a lifeline in the first minute of added time when Breel Embolo headed home from a deep free-kick and the same player should have grabbed an equaliser moments later, only to nod a golden chance over the bar at the far post.
Instead it was left to Werner to seal Leipzig's third win from three as he ran on to a long ball forward and finished low past Sommer.