The visitors dominated in midfield and the result duly went their way as the two new signings combined brilliantly with one-twos either side of half-time to seal the points
Both strikes were the products of a brilliant combination between the two new signings, each taking a turn to finish off a one-two with each other, though Steven Gerrard's lazy pass set up the counter for the German's opener.
After being caught in a tug-of-war between the two clubs over the summer, Real Madrid loanee Nuri Sahin made his debut for the hosts in the absence of injured-again midfielder Lucas Leiva, though there were no strikers available on the bench for Brendan Rodgers.
Theo Walcott, subject of speculation but still at the club after deadline day, dropped to the bench while Alex Oxlade-Chamberlain started in midfield for the Gunners and Vito Mannone continued in goal.
Arsenal saw plenty of the ball in the early stages but Liverpool threatened first, Fabio Borini finding space from 30 yards out to swipe wide. It was a sign of the visitors' high defensive line which was nearly exploited moments later when Luis Suarez shot past Per Mertesacker, who could only clumsily tug him back for a booking.
The Reds came close after 12 minutes when, thanks to Carl Jenkinson awkwardly clearing behind, Daniel Agger failed to get his head on Gerrard’s corner. The Dane had been totally unmarked and should have put his side ahead.
Arsenal had the better of possession in an otherwise even period, Suarez being cut off in the box by Mertesacker when he did get forward, as the midfield base of Mikel Arteta and Abou Diaby impressed.
The Spaniard was deservedly booked, though, when Raheem Sterling burst past him on a dangerous dribble, tripping him up when it was clear that the forward could not be caught. Gerrard's free kick, though, was wasted.
The captain was at fault, too, on the half hour when he lazily misplaced a pass in the Arsenal half. The Gunners brought the ball out on a swift counter, Cazorla racing through the middle before threading it through to Podolski, whose finish under pressure from the tracking-back Glen Johnson was as cool as it was accurate.
Liverpool, shocked, came close a few minutes later when the gambling Sterling stepped around a marker in the box and cut a high-difficulty shot against the outside of the post. The Anfield crowd appealed for a penalty when the 17-year-old went down in the box moments later but Mertesacker had got enough of the ball with a fine tackle.
Arsenal could have gone into the break two up if Olivier Giroud had not snatched at his effort lamely wide when played in by a neat pass down the inside-left channel, with the waiting Kieran Gibbs' unmarked status going unexploited. The visitors went into half-time deservedly leading, though.
Suarez tested Mertesacker five minutes after the break when he appealed for a push by the German in the box. There appeared to be some contact but the forward went down a little quickly; his reputation may have swayed the benefit of the doubt against him.
The pressure on the Uruguayan went up when Borini's withdrawal for Stewart Downing left him as the only remaining recognised striker for the Reds and he came close to rewarding that faith when Vito Mannone tipped over Liverpool’s first shot on target shortly before the hour mark.
Sterling continued to pose the most consistent threat for the hosts as they dominated proceedings in search of an equaliser, his jinking run and cross unrewarded with a missed header.
Cazorla came forward with a reminder of Arsenal's potency moments later, though, slicing a shot a whisker past the post as well as Reina’s outstretched hand, with the Gunners content to wait for opportunities on the counter.
The Spaniard got that opportunity in the 67th minute. It came from another fine combination with Podolski, the German this time taking his turn to play Cazorla in with a one-two. The import from Malaga picked it up and thrashed it off a helpless Reina into the net.
With Liverpool looking downcast and unlikely to overturn the visitors’ superiority, Gerrard was very lucky not to get at least a yellow card for a rasping challenge on Vermaelen, catching the Belgian’s leg when he had no hope of beating him to the ball. It should have made worse an already humbling showing from the veteran midfielder.
It seemed only a question of whether Arsenal would score any more and Giroud got in the hunt, being fended off well by Jose Enrique after nodding a set-piece over the bar from a decent position.
The Reds nearly got back in it with a late scramble, substitute Jonjo Shelvey forcing Mannone into a save, with Vermaelen on hand to smartly clear the rebound before the hosts could pounce. What further possession that Liverpool had was unpenetrative and Arsenal held on for their first league win of the season.