Werner has been persistently linked with a move to European champions Liverpool and it took the 24-year-old Germany international just eleven minutes to get on the scoresheet.
Yussuf Poulsen nodded in the second goal of the game on his 250th appearance for the club, with Marcel Sabitzer killing off the contest shortly before the break.
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Leipzig beat Mainz 8-0 earlier this season and although they did not score quite as many at the Opel Arena in west Germany, they did manage to add another two.
Werner was quick to pounce after the restart and he completed his hat-trick with a deft flick over Florian Mueller in the 75th minute, to become Leipzig’s all-time top goal scorer.
The win moves Leipzig back into third on 54 points, one place above Bayer Leverkusen and two above Borussia Moenchengladbach. The Bundesliga’s top two, Bayern Munich and Borussia Dortmund, clash on Tuesday evening.
Schalke dominated possession but could manage only two shots on target in the match and fell behind when Eduard Loewen curled home a sublime free kick midway through the first half.
Nigerian winger Noah Sarenren Bazee doubled the away team’s lead in the second half before Sergio Cordova wrapped up the victory in stoppage time.
Schalke, who have only scored two goals in their last nine games, are now without a win in nine and have slipped down to eighth place in the Bundesliga table. Augsburg meanwhile move up to twelfth, seven points above a relegation playoff spot.
Turkey international Kenan Karaman gave the away side the lead shortly before half-time, while Erik Thommy doubled Dusseldorf’s advantage in the second half.
But Anthony Modeste struck in the 88th minute before Jhon Cordoba’s dramatic stoppage time equaliser, to leave Dusseldorf in serious trouble towards the bottom of the table.
The Bundesliga became the first major sports league to resume last week, with no fans in attendance, after being suspended for more than two months due to the coronavirus pandemic.