Fernando Torres opened his account for the season as Liverpool beat West Bromwich Albion 1-0 on Sunday to claim its first win in the Premier League, while Darren Bent scored Sunderland's injury-time winner against Manchester City.
Liverpool started the match in the league's bottom three on just one point but Torres' 65th-minute volley allowed his team to climb into mid-table on four points. West Brom had James Morrison sent off with five minutes left.
Bent converted a penalty in the fourth minute of stoppage time as Sunderland won 1-0 against big-spending City, which was looking to join Arsenal and Manchester United on seven points.
Elsewhere, Aston Villa beat Everton 1-0 thanks to Luke Young's ninth-minute strike and Robbie Blake scored with a late free kick as 10-man Bolton recovered from going two goals down to draw 2-2 with Birmingham.
Chelsea leads the table with nine points after winning its first three games, the latest coming against Stoke on Saturday.
Torres started on his own up front after being rested for Liverpool's win in Trabzonspor on Thursday, which qualified the five-time European champions for the group stage of the Europa League, and played an integral part of most of his side's attacking moves at Anfield.
The Spain striker squandered a number of half-chances in the first half but found the target in the 65th minute when he met a cross by Dirk Kuyt with a right-foot volley into the bottom corner.
Center back Martin Skrtel also went close for Liverpool, who bounced back from a 3-0 defeat at City on Monday.
Morrison was sent off by referee Lee Probert after being penalized for a lunge at Torres as West Brom slipped to its second defeat in three matches.
A glaring miss by Carlos Tevez proved costly as City lost for the first time this season, dropping to ninth spot as a result.
Tevez missed an open goal in the 16th minute after being played in by Yaya Toure, and Bent made City pay by shooting his penalty low past Joe Hart deep into injury time after the England striker was fouled by Micah Richards.
Roberto Mancini's City side dominated the first half but failed to take advantage of their grip on the game, while Sunderland made the most of its opportunity despite losing striker Fraizer Campbell with what looked a serious injury.
Central defender Roger Johnson put Birmingham in front after four minutes against Bolton and was then involved in the incident that led to the sending-off of Bolton goalkeeper Jussi Jaaskelainen in the 37th.
Craig Gardner added a second for Birmingham in the 50th minute but Bolton bounced back, with Kevin Davies' penalty reducing the deficit and Blake's curling free kick 10 minutes from time sealing a superb comeback.
"I think anybody at that game would have struggled to recognize it was 11 against 10 as we were the team pushing on," Bolton manager Owen Coyle said. "We were certainly deserving of a point, minimum."