The Reds have struggled for form this season with their key players struggling.
Trent Alexander Arnold opened the scoring in the seventh minute with a sublime free kick from 25 yards out.
Rangers goalkeeper Allan McGregor was in an inspired form frustrating the likes of Darwin Nunez and Mohammed Salah.
Salah scored the second goal from the penalty spot after Leon King fouled Luis Diaz.
Trent Alexander-Arnold, who has been widely criticised for his poor performances this season praised his teammates for recovering from the poor performance against Brighton on Saturday.
"The performance was great from the lads. On Saturday it was disappointing.
"We never got going, our press wasn't there, we were slow out the blocks.
"But it's the complete opposite today.
"We started well and continued throughout the game.
"They had their spells and the second half was tough but overall we played outstanding.
"I don't normally score from that side. I usually score from the other side of the pitch.
"It was just about getting it on target, that's what I've been focusing on when I practice.
"If you hit the target, there's always a chance it goes in or you get a rebound."
In the other Group A match, Napoli spanked Ajax 6-1 at the Johan Cruyff Arena.
Up next for Liverpool is a trip to Scotland to face Rangers in the second leg.