Werner, who had struggled with a virus a few days ago, came on in the 63rd minute and volleyed in for the 2-1 lead six minutes later to take his league goal tally to nine.
The Germany forward, expected to be his nation's first choice at this summer's World Cup in Russia, then set up Bruma for their third in the 71st to help Schalke move up to 31 points, one above Schalke, who drop to third.
Leaders Bayern Munich picked up where they had left off before the winter break and beat Bayer Leverkusen 3-1 on Friday to stretch their advantage to 13 points on 44.
Schalke defender Naldo had earlier turned from villain to hero when he first deflected a Naby Keita shot into his own goal but then made amends with a header for a second half equaliser.
Leipzig, who had won none of their last four league games, missed a 37th minute penalty with Jean-Kevin Augustin but Keita fired in four minutes later.
Naldo levelled but Werner's entry turned the game around in favour of the hosts.
Hoffenheim had earlier missed their chance to move back to the top three after drawing 1-1 at lowly Werder Bremen.
Hoffenheim, who had lost three of their previous six league matches and were looking for a winning start to the new year, were in control after Benjamin Huebner put them ahead in the 39th minute.
But Theodor Gebre Selassie pulled Bremen level just past the hour mark, leaving Hoffenheim in eighth place on 27 points.
Eintracht Frankfurt and Freiburg also drew 1-1 while Augsburg moved on to 27 points with a 1-0 victory over Hamburg SV.
(Reporting by Karolos Grohmann; Editing by Christian Radnedge)
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