Luis Suárez will be back this week and suddenly Liverpool look like a team that need him. The suspended striker watched with a thoughtful expression as Southampton unexpectedly ruined Liverpool's attempt to make their best ever Premier League start, leaving Roberto Martínez's Everton as the top flight's only unbeaten side.

It was not supposed to work out this way, though if unexpected, Southampton's victory was certainly not undeserved. The visitors were the stronger, more inventive side throughout, and but for some goalkeeping heroics from Simon Mignolet could have won even more convincingly.

Mignolet's shot-stopping has been a significant part of Liverpool's rise to the top of the table, perhaps an indication that one day the club's luck would run out. This was that day. Brendan Rodgers tried everything, including a few things he will probably not be trying again, but nothing worked. Even presented with a more than passable opportunity to snatch a point at the death, when Daniel Sturridge played a pass over the Southampton defence to allow Raheem Sterling to run clear on goal, the substitute came up with a wretched first touch to practically pass the ball to Artur Boruc.

This will all have looked familiar to Suárez. All his side's forward options disappointed. Liverpool can only hope the one-man rescue act is still functioning as well as it has in previous seasons. "The only positive from today is that Luis's suspension is over," Rodgers said.

An undistinguished first half was enlivened only by a terrific save from Boruc, arching to his right to get one hand to a Steven Gerrard free-kick that was heading for the top corner. Dejan Lovren was lucky to get away with a challenge on Sturridge that looked like a penalty, and Boruc saved again from the elusive Victor Moses, but in terms of genuine goalmouth action that was it for the first 45 minutes.

Southampton, with Adam Lallana impressive down the right, were not afraid to come forward and threaten, though their attacks tended to founder on the rocks of Liverpool's uncompromising back row of four centre-halves. Rickie Lambert might be having a good season but, really, he's not that much of a handful. Kolo Touré was playing at right-back in the absence of the injured Glen Johnson, and Mamadou Sakho found himself pressed into service at left back despite his insistence on joining from Paris Saint-Germain that he preferred not to play in that position.

José Enrique was available but relegated to the bench, Rodgers evidently believing that a £16m player has to be accommodated somewhere. Touré managed to get up in support of the attack, as is his wont, whereas Sakho stayed back and deprived Liverpool of an outlet on the left.

Mignolet was called upon to save from Lallana at the opening of the second half, though only because the goalkeeper's attempt at a dribble on the edge of his area had got him into trouble. Rodgers values goalkeepers who can use their feet well; Anfield is debating whether Mignolet is a player who comes into that category or whether he is simply over-confident.

The Belgian was certainly well beaten when Southampton took the lead from a Lallana corner, needlessly given away through a mix-up between Touré and Martin Skrtel on the Liverpool right. Lovren lost Daniel Agger and produced an unorthodox but firm header, so much so that Gerrard, standing at the near post, could not reach the attempt in time to clear. Four centre-halves or not, Liverpool were undone at a set piece.

By the time Mignolet had saved again to deny Pablo Osvaldo the Liverpool manager had seen enough, and sent on José Enrique for Agger. That did not prevent Southampton pressing for a second, and only outstanding reactions from Mignolet kept them out when a great run from Luke Shaw left him looking certain to score, before the goalkeeper saved again to prevent Steven Davis following up.

It was not quite all Southampton – José Fonte had to clear off the line when Sterling crossed from the right – yet even when Lambert was withdrawn the visitors still looked the side most likely to score.

Southampton were actually the last team to beat Liverpool, at home towards the end of last season, and on this evidence they look good value for a top-half finish at the very least.

Not that Mauricio Pochettino is getting excited. "I have been in football too many years to get carried away," the Southampton manager said. "We won't be getting ahead of ourselves. We have just had an important win against the league leaders but tomorrow we will be back in training at 10am".