Yaya Toure and Sergio Aguero produced moments of brilliance in the final eight minutes to break Villa's stubborn resistance and provisionally move second-place City to within two points of Chelsea, which hosts Arsenal on Sunday.
"I have told the players that in May you are going to win or lose the Premier League by just a point or so, so every one you drop on the way makes it more difficult," City manager Manuel Pellegrini said.
"It is very difficult to recover the points against a team like Chelsea, and the only way to do it is to win every game."
Jordan Henderson set up Liverpool's opening goal by Adam Lallana, and grabbed the winner against West Brom, lifting the anxiety around Anfield after the club's run of three league games without victory.
"We needed to find a way to win today," said Liverpool manager Brendan Rodgers, whose team has struggled to maintain the form of last season that swept it to a second-place finish behind City.
Liverpool's problems are nothing compared to those of Newcastle, which remained winless this season after a 2-2 draw at Swansea.
The pressure on Newcastle manager Alan Pardew would have been fierce with another defeat but he was helped out by two equalizers from Papiss Cisse, who also scored twice to snatch a draw against Hull two weeks ago.
"Anybody who saw the game today will see the players gave everything," Pardew said.
"It's my job to try and lift the players and make sure there's fight in their belly. The confidence will get better and this will give us belief."
Sunderland beat Stoke 3-1, Hull defeated Crystal Palace 2-0 and Burnley scored through a 90th-minute free kick to draw 2-2 at fellow promoted side Leicester.
Toure has been criticized by some pundits for failing to keep up his high standards from last season, when he scored 24 goals for City in all competitions.
But the Ivory Coast midfielder showed his importance to the champions by finally finding a way past Villa's American goalkeeper, Brad Guzan, who had produced a string of saves in a one-sided second half and also seen his goal frame struck by Aguero and Aleksandar Kolarov.
"I'm not a machine — I'm a man," said Toure, who dribbled between two players and calmly stroked in a left-footed finish for his goal in the 82nd minute.
Aguero ensured there was no way back for Villa by collecting a pass just outside the area, giving himself space and smashing a low shot into the corner in the 88th.
The memory of Liverpool chasing City in last season's title run-in is fading after the Reds' underwhelming start to this campaign, with their only previous victories coming in August.
The performance against West Brom was unconvincing, too, with Daniel Sturridge still out injured and Mario Balotelli dropped to the bench. Crucial interventions from Henderson made the difference, though.
The England midfielder's clever flick played in Lallana, who converted his first goal for Liverpool to give the hosts the lead in the 45th.
Saido Berahino equalized with a 56th-minute penalty the forward won from a challenge by Dejan Lovren that appeared to be made outside the area, but Henderson restored Liverpool's lead with a goal that lifted the team into the top half of the standings.
"It's been frustrating," Rodgers said. "We had to make changes in the summer to strengthen the squad and we missed a world-class striker (Luis Suarez). We will get better as we go on."
Pardew's unstable position has been one of the talking points of the first two months, and the draw at Swansea should keep him in place over the upcoming international break.
The sinking feeling for Newcastle's disheartened fans returned when Wilfried Bony put Swansea ahead in the 17th but Cisse equalized by getting a touch on Gabriel Obertan's cross to the near post.
Gylfi Sigurdsson picked up another assist when he laid in Wayne Routledge for Swansea's second goal in the 51st, only for Cisse to rescue Newcastle with 15 minutes left.