The vice-captain calmly converted a spot-kick nine minutes from time after a foul on Sadio Mane, whose first-half goal was cancelled out when former Reds forward Ryan Babel pounced on a defensive mix-up.
It meant Jürgen Klopp’s side moved back to the top of the table, two points clear of Manchester City having played one match more than the reigning champions.
Team news
Klopp made two changes to the XI that started in Munich in midweek, with Fabinho and Adam Lallana selected to replace Jordan Henderson, who missed out due to injury, and Milner.