The home side looked threatening from the outset and Babel capitalised in the 11th minute when Cenk Tosun cut in from the left and crossed for the Dutch winger to fire a side-footed shot beyond Leipzig goalkeeper Peter Gulacsi.
Talisca doubled the Istanbul side's lead in the 43rd minute, meeting a cross from Ricardo Quaresma with a diving header from close range.
The Bundesliga side made a brighter start to the second half but their momentum was interrupted when a floodlight failure brought proceedings to a halt for 10 minutes.
Upon resumption Besiktas were content to soak up the pressure and Leipzig were unable to find a way through in their second ever game in the Champions League.
Besiktas lead Group G with six points while Leipzig are level on one point with last season's semi-finalists AS Monaco who were beaten 3-0 at home to second-placed Porto.
(Writing by Daren Butler, editing by Tom Hayward)
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