Salah put the Premier League champions ahead in the first minute, latching on to a poor backpass and bursting goalwards before slipping the ball past goalkeeper Jesper Hansen.
The goal was Salah’s 22nd in the Champions League making him the club’s record scorer in the competition.
The Danes drew level in the 62nd minute through an Alexander Scholz penalty after Liverpool keeper Caoimhin Kelleher brought down Anders Dreyer. The linesman had initially flagged for offside but that decision was overturned by VAR.
Scholz had an effort ruled out for offside in the 77th minute while Liverpool’s Japanese midfielder Takumi Minamino also had the ball in the net but the goal was not awarded following a VAR review.
Jurgen Klopp had fielded a weakened and youthful team in the game against the bottom club in the group.