James Milner scored an early goal but Daniel Sturridge had to score two as the reds grabbed the three points despite defensive lapses.
Liverpool made only one change from last week's Premier League outing with the injured Christian Benteke being replaced by Danny Ings - a man with two goals in two games for the reds.
There was immediate delight for those around the ground when in the second minute of the game Coutinho found Milner at the edge of the box - the captain then firing a left footed shot into the corner of the net and putting the reds 1-0 ahead.
A fist pump from the Liverpool manager and huge relief for the expectant crowd inside the ground.
Milner continued to have an important presence - often finding Moreno who was playing as far forward as we have seen of a Liverpool wing-back this season. At every opportunity the young Spaniard looked to find Sturridge or Ings in the middle.
Liverpool had the bulk of the possession in the first half - enjoyed a number of corners - but with Villa mounting a rearguard action they were unable to break Villa down further - Coutinho trying his luck at long distance shots.
As the half wore on Liverpool lost their confidence.
An error from Can presented a chance to Gestede who snatched at his shot. Later on Mignolet palmed a cross into the danger zone and Liverpool were lucky to hold their lead into the half time break.
The second half and Liverpool needed to get that second to give them a cushion.
It duly arrived from Daniel Sturridge who had looked poor in the first half. Just before the hour his short pass found Milner, who flicked a delightful chip over the Villa defence - allowing Sturridge to fire a left footed volley past Brad Guzan. 2-0.
Or maybe not.
At the other end a Villa break and a cross from near the touchline was sent right across the face of goal - Gestede presented with a simple tap in to reapply the pressure.
But Sturridge had another goal in his locker - this time with a one-two with Coutinho. The Brazilian's back-heel setting up Sturridge perfectly who controlled with his left before firing in with his right.
Almost immediately Villa, through Gestede grabbed their second - Gestede heading in to give Rodgers more headaches about Liverpool's struggle with defending.
Late on Sturridge and Ings made a break, but Ings failed to get the goal his effort possibly deserved. Sturridge too missed out on the chance to get a hat-trick opting to move the ball to his left foot when an early right footed shot may have done the damage.
Liverpool get an important win but the reds are far from the finished article. Two more wins are needed in the week ahead - a midweek home game against FC Sion in the Europa League before a weekend trip across Stanley Park to face local rivals Everton.
Sturridge's first goal