Victory lifted Leipzig, last season's Bundesliga runners-up, to six points and provisionally into third place, ahead of their opponents on goal difference.
Leipzig had a good chance in the first half but Jean-Kevin Augustin's effort bounced off the post as they took the initiative from the start.
Hamburg were then saved by the video assistant referee on the stroke of halftime when Werner was brought down in the box and the referee initially pointed to the penalty spot before changing his mind after asking for a video replay.
The hosts, looking to make it three wins in a row at the start of a season for the first time since the 1974/75 campaign, again survived early Leipzig pressure in the second half.
But keeper Christian Mathenia could do nothing when Keita, who will join Liverpool next season, thundered in an unstoppable shot past the keeper in the 67th minute.
Germany international Werner then showed why he is the most in-form striker at this moment, charging through and finishing his solo effort seconds after Hamburg missed a golden chance to level in the 75th.
Champions Bayern Munich will look to make it three wins in a row when they travel to Hoffenheim on Saturday. Leaders Borussia Dortmund face Freiburg.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar)
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