The win over Villa gave the Reds a chance to force their way into the top four, and they did just that.
The Reds recorded a third successive league win for the first time this season and narrowed the gap to the top four to five points.
The perfect start to the game was created by a couple of passes from Trent Alexander-Arnold and Andrew Robertson, and they were able to take the lead in the fifth minute.
The visitors were able to add a second before the end of the first half when Van Dijk's shot went in off Konsa.
Unai Emery's side were rewarded for their persistence after the restart when Watkins headed in Douglas Luiz's cross, after Villa had missed a couple of good chances.
Home hopes of a dramatic fightback at Villa Park were ended by Bajcetic's goal nine minutes from time, which followed great work by Nunez to recover the ball at the byeline.
He said he was pleased that his side would resume their campaign within punching distance of the top four, but he felt it was still a long arm.
With a game in hand on fourth-placed Spurs, the boss must now feel confident of reaching their goal.
Spurs drew with Brentford earlier in the day and their first-half double put the Reds in a good position.
Robertson's crucial role in the opener - setting up Salah with a first-time ball across the six-yard box after Alexander-Arnold's stunning outside-of-the-foot pass - saw him overtake LeightonBaines as the defender with the most assists in the competition's
It was not a job done for the Reds against determined opposition and they were relieved to add a late third after weathering intense Villa pressure.
After dropping 20 points from their first 14 games, there are signs of life for last season's runners-up.
There's more to come.