Big-spending Manchester City, who beat Liverpool 3-0 on Monday, lost 1-0 at Sunderland where Darren Bent's stoppage-time penalty gave the home side all three points.
Bolton Wanderers fought back from 2-0 down to draw 2-2 at home with Birmingham City after their Finnish international goalkeeper Jussi Jaaskelainen was sent off for slapping Birmingham's Roger Johnson in the face.
Torres had two knee operations at the end of last season, struggled at the World Cup for Spain and, according to manager Roy Hodgson, is still not quite at full fitness. Nevertheless, he volleyed in Dirk Kuyt's cross after 65 minutes for his 41st goal in 43 Premier League matches at Anfield.
"Fernando is fighting to be at his very best and is still two to three games away from full match fitness but the goal will do him a power of good today," Hodgson told the BBC. "He got stronger as the game went on."
Manchester City's Carlos Tevez was guilty of a glaring miss after 14 minutes when he shot over the bar from 12 metres out with an open goal in front of him.
The deadlock with Sunderland was finally broken in stoppage time when Micah Richards fouled Bent to concede the penalty and Bent struck his spot-kick underneath England's current number one goalkeeper Joe Hart to score.
Sunderland boss Steve Bruce told BBC Sport: "I don't think Carlos Tevez will ever miss a chance like that again. It just shows you what football is. Thankfully a little bit of luck went our way but in the second half I thought we were the only side who would win it."
Villa's victory moved them into fourth place in the table with six points from three matches, behind Chelsea (9 points), Arsenal (7) and Manchester United (7) who all won on Saturday.
