A fitting end to a dreadful few days in the Liverpool reign of Brendan Rodgers stemmed from a superior Arsenal display, carelessness and another error from the once formidable José Reina. But it was strikers in absentia who hogged the limelight. The Liverpool manager admitted he would never have loaned Andy Carroll to West Ham United had he foreseen Friday's non-transfer deadline day and will consider an Anfield return for Michael Owen.

Arsène Wenger appeared mildly irritated as he entered the Anfield press room having been made to wait an age to review Arsenal's first win of the season by Rodgers's inquest into Liverpool's shambolic end to the transfer window. Wenger was lucky he was not kept waiting all night. A rift has opened between the Liverpool manager and the club's owners, Fenway Sports Group, after they refused to offer more than £4m to Fulham for Clint Dempsey 24 hours after Carroll's exit and Rodgers's diplomacy could not disguise it.

Asked if he would have sanctioned Carroll's move knowing what was to come on Friday, Rodgers said only: "No." Asked how confident he was that a replacement was coming in when he allowed Carroll to go, he said: "Very." It was the admission that Owen is under consideration eight years after leaving Anfield and four months after being released by Manchester United that showed the desperation that now prevails.

"Any player that I believe can improve the squad I will look at. No question," added Rodgers. "We have a very small group – 19 players trained [on Saturday], with young players amongst that. I cannot say no." Owen has been offered a two-year contract worth a basic £1.5m a year by Stoke and can sign at any time as a free agent. Didier Drogba is not an option at presentas his contract with Shanghai Shenhua has not been cancelled.

How Arsenal revelled in Anfield's problems. Off the pitch, their supporters chanted "Andy Carroll, he would have scored that" as Jonjo Shelvey squandered a late chance. On it, the summer signings Lukas Podolski and Santi Cazorla scored Arsenal's first league goals of the season while Theo Walcott, a player Rodgers had hoped to sign and will make a renewed attempt for in January, sat on the bench throughout.

Liverpool had their moments and three penalty claims denied by the referee, Howard Webb, but otherwise no complaints at a merited defeat. This may not be the free-flowing, expansive Arsenal side of old but their midfield trio of Abou Diaby, Mikel Arteta and Santi Cazorla dominated their opponents and a third clean sheet of the season supported Wenger's conviction that the potential for vast improvement exists. Rodgers, by contrast, has yet to record a league win as Liverpool manager and this is the club's worst start after three games since 1962.

Whatever the ramifications from Liverpool's non-event of a transfer deadline day, Rodgers started with a potent lineup improved from last Sunday's draw with Manchester City by the return from suspension of Daniel Agger and the full debut of Nuri Sahin. Fabio Borini put efforts high and wide, Agger should have done better with a free header from a Steven Gerrard corner and Raheem Sterling spun and shot against the outside of the post after a knockdown from the Liverpool captain.

Gerrard's attempt at a first-time pass to Luis Suárez deep into Arsenal territory was to have damaging consequences, however. Podolski intercepted it and played a good ball through to Cazorla before sprinting half the length of the pitch for the return. Cazorla, calmness and awareness personified, delivered the return Podolski's effort deserved and the German international buried Arsenal's first league goal under Reina into the far corner of the goal in front of the Kop.

Olivier Giroud then wasted a clear chance to open his Arsenal account before Cazorla put the game beyond Liverpool. Again it was easily avoidable. The Spain international compounded a bad week for Liverpool and Reina – guilty of a poor error against Hearts in midweek – when he exchanged passes with Podolski and tried his luck with a shot hard and low. The ball flew under his compatriot's grasp and in.

Rodgers said: "The goals we conceded had nothing to do with systems or style, or the tactical element. We gave the ball away for the first goal and didn't do well enough to stop the shot. The second goal, we had enough bodies behind the ball to defend the goal. These are the hard yards you have to put in early on but I wouldn't change it for anything."

Liverpool had appealed loud but in vain for second half penalties when Per Mertesacker wrestled with Suárez and when the ball hit Thomas Vermaelen's arm, but rarely threatened a recovery. It was, in the grand scheme of things, the least of Rodgers's concerns.

As he explained: "It is a learning process for the owners as well. They have come in and invested well over £100m and then they have made the change for whatever reason and one of the most iconic figures in the club's history has left. They made the change and made a big commitment to have me here for the longer term.

"I have a group of people I worked well with and the owners have been very up front and honest with me. I have no problem with that. There are obviously one or two things we need to iron out but the owners have been very honest and haven't misled me in any way whatsoever. There are operational things we need to sort out and if we do that, that will help us in the next transfer window.

"I have spoken to people back in America since Friday. They have my thoughts. We move on and will reflect on it next week. The reality is not a lot is done in January but we certainly need help."

He would not expand any further. "It's gone," said Rodgers on the issue of Carroll, who can be recalled by Liverpool in January. The optimism and tranquillity he had hoped to find at Anfield has vanished too.

Man of the match Abou Diaby (Arsenal)