An Ian Rush header gave Liverpool their first away victory since mid-December against QPR and the Merseysiders could even afford the luxury of two missed penalties.
The Welsh international striker - dropped recently after managing only three league goals all season - has now responded with two goals in as many games.
Steve McManaman's 72nd minute cross did the damage - Rangers' goalkeeper Tony Roberts failing to cut it out allowing Rush to pounce with a decisive far-post header.
It was cruel luck for Roberts, who had been his side's hero after Liverpool had won a penalty after the goalkeeper's 57th minute challenge on Rush. Mike Marsh struck the spot-kick firmly enough but Roberts dived to his left to turn the ball wide.
There was more penalty drama to come when Clive Wilson clearly pushed Don Hutchison in the box six minutes from time. This time, skipper John Barnes tried his luck from the spot but drove the ball wide.
Barnes had provided Liverpool's main first-half threat, hitting the crossbar with a ferocious 22nd minute drive and curling another shot just over from the rebound.
Rangers went close after just eight minutes, Devon White rising to meet Wilson's corner with a powerful header which Liverpool goalkeeper David james gathered at the second attempt.
After 25 minutes, White exchanged passes with Wilson, surged into the box but shot straight at James. The 'keeper allowed the ball to bounce away from him but redeemed himself by gathering bravely at the feet of Les Ferdinand.
After the break, it was virtually one-way traffic for Liverpool with Roberts doing well to save one rising Don Hutchison shot.
Liverpool striker Ian Rush claimed that he had regained his "belief and confidence. It's nice to score. I think that in the position we are in, it was more important that we got the three points," said the striker. "The way we played tonight it would have been a travesty if we hadn't got the three points."
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