The win meant that Liverpool had won consecutive league games for the first time since December 2011 and they started the game well by dominating possession but shots on target were hard to create as it took until the 15th minute before either side had a real chance.
Coutinho curled the ball over the bar after failing to test Guzan but six minutes later, a clever move saw Coutinho dummy a pass from Enrique which found its way to Sturridge who showed neat footwork before smashing the ball past Guzan to the joy of the travelling fans.
Aside from taking the lead, the strike was the 100th goal of Rodgers' reign as Liverpool manager and Liverpool were in complete control but that was threatened just before the break as Mignolet was called into action to deny a curler from Benteke and did so with a full length save to protect the lead into the interval.
Villa pressured the visitors as Bacuna hit a dangerous free-kick which found the Liverpool wall and within seconds, Westwood drove a shot over the bar. Liverpool's new loan signing, Aly Cissokho came on in place of Iago Aspas for his debut and Villa continued to threaten an equaliser which they nearly had when Benteke rose to meet a corner.
The contest entered the final stages and Tonev forced a low save from Mignolet but Benteke connected to a bouncing ball into the box and hit a powerful effort which saw Mignolet palm the ball past the post and make sure that he claimed a second consecutive clean sheet and three points that put Liverpool up to joint top on six points.