Keita broke the deadlock four minutes before halftime when he collected the ball just inside the Bremen half, took advantage of a lucky bounce, slalomed his way past several defenders and rounded Felix Wiedwald to score.
Wiedwald made a superb double save to deny Emil Forsberg and Yussuf Poulsen before Keita struck again in the 75th minute, the Guinea midfielder rifling the ball home after being set up by substitute Davie Selke.
Serge Gnabry turned in a cross from Fin Bartels to pull one back for Bremen one minute later and set up a dramatic finale.
Wiedwald was sent up for a corner as Bremen pressed for an equaliser in stoppage time but the gamble backfired as Leipzig cleared the ball, broke quickly and Selke rolled the ball into an empty net.
Leipzig, only founded in 2009, remain unbeaten after eight games and have 18 points, two behind leaders Bayern Munich who beat Borussia Moenchengladbach 2-0 on Saturday.
Hertha, who beat Cologne 2-1 on Saturday, are a further point behind in third with Hoffenheim fourth on 16 points.
Nabil Bentaleb became the first Algerian to score two goals in a Bundesliga match as he helped Schalke 04 continue their revival with a 3-0 win over Mainz 05.
Max Meyer was also on target for the Royal Blues who lost their opening five games but have taken seven points from their last three. Schalke moved up to 14th place, ahead of Bremen on goal difference.
(Writing by Brian Homewood; Editing by Ian Chadband and Pritha Sarkar)
