Liverpool's victory on Saturday came courtesy of an Aaron Ramsey own goal and a last-minute strike from Luis Suarez, but the defeat could have greater ramifications for Arsene Wenger's side ahead of next week's crucial Champions League play-off second leg in Udinese and the small matter of a trip to Manchester United.
Samir Nasri - whose £22million move to City is, according to reports on French TV, said to have broken down - did not applaud the crowd as the teams came out of the tunnel, and was then jeered by some sections of the Emirates Stadium faithful when his name was announced.
Liverpool showed early intent, as Dirk Kuyt - who netted a late penalty deep into stoppage time to earn a 1-1 draw here last season - drilled a low, angled drive goalwards.
New Arsenal captain Robin van Persie looked lively as he nicked the ball out on the right channel and floated over a cross which Martin Kelly stabbed behind for a corner.
On eight minutes, Frimpong was cautioned for squaring up to Jordan Henderson as the Liverpool midfielder tried to take a quick throw-in.
Koscielny added to Wenger's defensive problems when he pulled up with what looked like a lower back injury on 14 minutes and had to be replaced by youngster Ignasi Miquel, the Spaniard making his Premier League debut.
Liverpool almost took the lead on 20 minutes when Andy Carroll's towering header at the far post brought a brilliant save from Wojciech Szczesny.
The visitors were in full flight now, with Henderson sending a tame header straight at the Arsenal keeper following another quick break down the left.
Charlie Adam then tried an audacious chip from the halfway line, which had Szczesny backpedalling, but flew over the bar.
Arsenal finally mounted a meaningful attack after 30 minutes when Frimpong's low drive from 20 yards was pushed wide by Jose Reina.
Nasri then showed what Arsenal would be missing when he picked up the ball in his own half, dashed forwards and turned Liverpool inside out before unleashing a 25-yard strike which curled past the post.
As the heavens opened for the start of the second half, Liverpool continued to frustrate Arsenal.
The visitors broke quickly into the Gunners box on 55 minutes, with Carroll laying the ball back to Kelly - and his snapshot struck the outside of the post.
Arsenal eventually started to up the pressure, but lacked a decisive pass in the final third.
Andrey Arshavin turned Kelly and fed Van Persie at the near post, but his goalbound flick was kicked behind by Reina.
With 20 minutes left, Arsenal were reduced to 10 men when Frimpong was shown a second yellow card when he left his foot up in a 50/50 with Lucas, right in front of the referee.
Liverpool snatched a fortunate lead on 78 minutes when as Suarez darted into the six-yard box, Miquel's clearance cannoned off Ramsey's chest and back over the keeper into the net.
A miserable afternoon for the Gunners was compounded in the final minute when Suarez tapped in from close range following a quick counter-attack.