Liverpool eased the burden on Luis Suárez with perfect timing. The Uruguay striker has carried the threat for Brendan Rodgers's team almost single-handed this season but his brilliance was not required against Southampton as Daniel Agger secured back-to-back league wins at Anfield for the first time since September 2011. They will have to repeat the feat at West Ham at the weekend when Suárez will be suspended following a booking here for handball.

The Liverpool manager bristled at a suggestion that Suárez's fifth booking arrived from an act of gamesmanship. In fairness to Rodgers, he probably did not see controversy coming on an otherwise comfortable afternoon. Not since Bolton and Wolves were despatched early last season have Liverpool enjoyed successive league wins at home. An end to that dreadful statistic was rarely in doubt against a limited Southampton side.

"The game should have been put to bed in the first half," Rodgers said. "We were outstanding in the first half. Our tempo, passing and movement was excellent. We just didn't have any luck with the final ball. We had to defend for the last 10 minutes at 1-0 but there is no question we should have been out of sight."

Rickie Lambert was intent on making an impact against his boyhood club and his shoot-on-sight policy commenced early with a 25-yard effort wide of José Reina's goal. But that was a rare moment of adventure from Southampton who, for all Nigel Adkins's appeals to Jason Puncheon and Gaston Ramírez, did not threaten again until Lambert almost caught out Reina from 35 yards in first-half stoppage time. The performance proved a marked contrast to Southampton's start to the season, with their defending at Anfield more impressive than their ambitions in attack.

Credit for that belonged to Liverpool and their dominance in possession. Patience was also important as the hosts created few clear-cut openings before Agger headed them ahead. "We usually have a lot of possession but Liverpool had a better grip of the ball than we did and their movement was very good," Adkins said.

The return of Lucas Leiva from a torn thigh muscle sustained in August instantly improved the balance in Liverpool's midfield, with the Brazil international allowing Steven Gerrard and Joe Allen to push on from the right and left respectively and to keep Southampton encamped deep inside their own half. "It was great to have Lucas back," Rodgers said. "He is important to what we are trying to do here. He offers the team great stability and he can deal with the ball."

Liverpool found their rhythm midway through the first half and Gerrard, twice, Suárez, Glen Johnson and Jonjo Shelvey all threatened before the home side's best opportunity arrived via a route-one clearance from Reina. The ball hit Southampton defender José Fonte on the head, Suárez nipped in and turned the ball inside Paulo Gazzaniga, but Maya Yoshida intercepted to deny the striker a simple finish.

Shelvey struck the inside of the post from a Gerrard pass. Suárez picked himself up after a foul from his international colleague Ramírez and curled the free-kick against the bar. Johnson was first to react, retrieving possession and cutting back on to his left foot to float a delightful cross towards Agger, who steered a towering header into the top corner. It was no more than Liverpool deserved.

José Enrique twice went close to extending the home side's advantage after the interval, when Liverpool's display dipped – mainly due to consistently sloppy passing from Raheem Sterling and Shelvey – but Southampton lacked the quality to capitalise.

The only incident of note came 10 minutes from time when Suárez attempted to divert Gerrard's cross into the Southampton net with his hands. The man who denied Ghana a place in the 2010 World Cup semi-finals with a deliberate handball collected his fifth booking of the season and will miss the trip to Upton Park on Sunday as a consequence.

"I didn't see it to be honest," Rodgers said. "I knew he was booked but I didn't know what for." Asked if he thought deliberate handball constituted cheating by Suárez, the riled Liverpool manager added: "I don't think it is cheating, no. Cheating? No."