Liverpool headed into the game on a poor run of form, which has already all-but ended their Premier League title hopes.
In the Champions League, meanwhile, they were smashed by Napoli and scraped a win over Ajax.
However, Rangers had lost two and not scored a goal in the Champions League this season before Tuesday, meaning Liverpool were still favourites.
And, when all was said and done at Anfield, Liverpool came away with a comfortable win.
Liverpool opted to start in a 4-2-3-1 following their poor form, with Diogo Jota playing behind Darwin Nunez.
Nunez started well and won a free-kick inside seven minutes, which Trent Alexander-Arnold scored.
Liverpool continued to dominate and probably felt somewhat hard done by the fact they did not score again in the first half, but Alan McGregor was in good form.
The home side then started brightest in the second 45 and won a penalty after 51 minutes through Luis Diaz, which Mohamed Salah scored.
Rangers had a few good chances late in the tie but were unable to get a goal back.
Liverpool player ratings
Alisson (5) - Had very little to do for much of the game.
Trent Alexander-Arnold (6) - Great free-kick to open the scoring. Was once again defensively suspect, though, being turned by Kent several times.
Joel Matip (6) - A solid showing, playing out from the back and driving forward to help start attacks.
Virgil van Dijk (5.5) - Decent display but had little to do.
Kostas Tsimikas (5) - His crossing could have been better. Made a goal-line clearance late on.
Jordan Henderson (6) - Played a couple of good balls to start attacks and had little to do defensively.
Thiago (5.5) - Controlled the game well with his quick passing. Was lapse defensively, which led to a few Rangers’ chances.
Mohamed Salah (6) - Salah made some good passes to find Nunez and tested McGregor with his curling efforts from the right. Scored from the spot.
Diogo Jota (6) - Caused Rangers issues with his free-roaming role behind Nunez.
Luis Diaz (6.5) - Diaz made some decent chances for himself, cutting inside and curling efforts at goal. He also won a penalty, which Salah scored.
Darwin Nunez (6) - Started brightly, getting a shot off and drawing a free-kick, which TAA scored. Tested McGregor a few times but was unable to score.
Roberto Firmino (4) - Did little of note.
Fabinho (4.5) - Helped Liverpool see out the game.
James Milner (N/A)
Joe Gomez (N/A)
Rangers player ratings
Allan McGregor (7) - Could not do much about the opener. Made several good saves to deny Nunez in the first half.
James Tavernier (4) - Offered little going forward and did a poor job of marking Diaz.
Connor Goldson (4) - Had a tough time dealing with Nunez.
Leon King (2) - Was unable to deal with the pace of Luis Diaz, giving away a penalty in the second half.
Boran Barisic (3) - Allowed Salah to cut inside on his left far too much.
Malik Tilman (4) - Offered little at either end before being subbed off.
John Lundstram (4) - Struggled to mark Diogo Jota.
Steven Davies (4) - Spent much of the game in his own half and did not win many duels, often being by-passed by balls from Thiago and Henderson.
Ryan Kent (5) - Did not see too much of the ball but caused Trent problems throughout the night.
Alfredo Morelos (4) - Was very isolated up top and did not test Alisson.
Ryan Jack (4.5) - Caused some issues for Liverpool with his pace but did little of note.
Fashion Sakala (N/A)
Antonio Colak - Was brought on to add some attacking threat but missed a great chance 85 minutes in.
Glen Kamara (N/A)
Rabbi Motonda (5) - Would have scored if not for a great clearance by Tsimikas.