Werner joined Chelsea in the close season and missed their run to their Champions League semifinals in August but Leipzig had little trouble finding the back of the net against Mainz, striking twice in four minutes.
With Mainz struggling to find ways of stopping Leipzig, Forsberg almost scored again on the half hour mark when he dribbled past two opponents but his clever flick bounced off the post.
Mainz cut the deficit three minutes after the restart when Quaison threaded a ball through the entire Leipzig defence for Jean-Philippe Mateta to score with a fine finish.
Any hopes of a comeback lasted only three minutes, however, with Amadou Haidara tapping in a Forsberg assist in the 51st minute to restore Leipzig's two-goal cushion.
"On the first matchday it is all about starting the season well," said Leipzig coach Julian Nagelsmann. "It is important to live up to your own expectations. We wanted to show from the start that we are fully there."