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, surpassing Steven Gerrard.
"Exceptional player, absolutely exceptional player. Since we worked together obviously a lot of things clicked really for all of us. He helps the team massively and he knows and appreciates the help of the team as well," Liverpool manager Jurgen Klopp said about Salah's achievement.
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.
Klopp had fielded a weakened and youthful team in the game against the bottom club in the group, giving Leighton Clarkson and Billy Koumetio their Champions League debuts.
"Tonight they helped us a lot because we could rest other players which was very important and there was actually no alternative. So yeah, I'm pretty sure it was a big night for both of them and I am really thankful that they are already as good as they are," said Klopp.