The England striker, 27, scored 15 minutes after coming off the bench by chipping in the Reds' third goal.
But he immediately pulled up and was substituted on 87 minutes, with boss Jurgen Klopp saying he hoped "it's nothing serious".
Sadio Mane and Mohamed Salah also scored as Liverpool set up a final against Atletico Madrid on Wednesday.
Mane slotted home the opener on six minutes after a quick breakaway by Roberto Firmino.
The Senegal forward then turned provider when he flicked the ball to Alberto Moreno, and although his effort was saved, Salah was able to nod in from close range.
Bayern offered little in attack but were in their most dominant spell when they were caught on the counter-attack and Sturridge raced forward to add a third before limping off.
Injuries have limited Sturridge to 46 league appearances in the past three seasons.
Speaking after the match, Klopp said: "He hurt his thigh. He has really prepared well, really well, so I hope this is just a small injury and that he can play again soon.
"The goal was great and he had a chance just before that. You could see how fast he is."
Liverpool, who remain unbeaten during their pre-season campaign, will play Atletico Madrid in Munich on Wednesday (19:30 BST) before their final friendly of the summer against Athletic Bilbao in Dublin on Saturday.
Daniel Sturridge went off with a thigh injury after scoring in Liverpool's 3-0 win over Bayern Munich in the Audi Cup.