The Germany international found the back of the net each side of the interval to put Salzburg on four points from two games, two ahead of Sevilla.
French champions Lille, who reduced the arrears through Burak Yilmaz, are bottom with one point, one behind Wolfsburg who drew 1-1 home draw against Sevilla.
The 19-year-old Adeyemi won his fifth penalty in four competitive games when he was brought down by Sven Botman and the speedy forward converted the spot-kick to put the hosts ahead in the 35th minute.
Salzburg got another penalty -- their fifth in two Champions League games -- when Yilmaz handled the ball from Mohamed Camara's free kick and Adeyemi buried it eight minutes into the second half to double the advantage.
Keeper Philip Koehn was at fault as Yilmaz scored with a free kick after 62 minutes to give Lille hope but Salzburg dominated the closing stages.